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Vatican City (CNN) -- Pope Francis on Thursday emphasized church advancement in his first Mass with the cardinals who elected him pontiff a day earlier.
With solemnity, he delivered a homily about moving the Catholic Church forward to the cardinal electors, who were dressed in light yellow robes. Altar servers burned incense in the Sistine Chapel, the setting for the Mass.
Speaking in Italian, Francis didn't use a script and kept the sermon short, calling on the cardinals to have courage.
"When we don't walk, we are stuck. When we don't build on the rock, what happens? It's what happens to children when they build a sand castle and it all then falls down," the new pontiff said.
"When we walk without the cross, when we build without the cross and when we confess without the cross, we are not disciples of Christ. We are mundane," he said. "We are all but disciples of our Lord.
"I would like for all of us, after these days of grace, that we find courage to walk in the presence of God ... and to build the church with the blood of Christ," the pope continued. "Only this way will the church move forward."
During the service, the cardinals prayed for the new pope and Pope Emeritus Benedict XVI so "that he may serve the Church while hidden to the world, in a life dedicated to prayer and meditation," the Vatican said.
When Jorge Bergoglio stepped onto the balcony at the Vatican on Wednesday evening to reveal himself as the new leader of the world's 1.2 billion Catholics, he made history as the first non-European pope of the modern era, the first from Latin America, the first Jesuit and the first to assume the name Francis.
Francis began Thursday by praying at the Basilica of Santa Maria Maggiore in Rome, a place of special significance for the Jesuits.
His next public appearance is likely to be Sunday. The new pontiff will "very probably" celebrate Mass at St. Peter's and then deliver the traditional Angelus blessing, said the Rev. Federico Lombardi, a Vatican spokesman.
But it won't be until Tuesday that Francis will be formally installed as pope.
That's by design. The day coincides with the Feast of St. Joseph, the patron saint of Italy.
In a letter dated Wednesday to Rome's chief rabbi, the new pope promised "renovated cooperation" between Catholics and Jews.
"I vividly hope I'll contribute to the progress" of relations between Jewish and Catholic people that they "have known starting from the Vatican II Council" in the 1960s, Francis wrote to Riccardo Di Segni.
The new pope said he was also acting in a spirit of "helping the world to be always more in harmony with the will of the Creator."
The new pontiff will meet with all the cardinals, not just those who were eligible to vote for him, on Friday and will hold an audience with the media on Saturday, Lombardi said.
Already, a picture is emerging of a humble man who shies away from the trappings of his new status and is devoted to his pastoral duties.
As pope, Francis will have plenty to deal with. He takes the helm of a Roman Catholic Church that has been rocked in recent years by sex abuse by priests, and claims of corruption and infighting among the church hierarchy.
Reflecting the urgency of those concerns, a group representing the alleged victims of sexual abuse by priests has written an open letter to Francis requesting a meeting.
"Your predecessor met only a few times with a few carefully chosen victims in tightly choreographed settings, as he visited nations where this crisis had reached a fever pitch," the letter from the Survivors Network of those Abused by Priests states.
"We write today seeking a different kind of meeting -- one in which our respective organizations -- yours, huge and struggling, and ours, small and struggling -- can begin to work together to safeguard children across the globe."
Meanwhile, in Los Angeles, one alleged victim of priest sex abuse, Michael Duran, urged Pope Francis to give Catholics new hope and make priests and cardinals accountable for their actions in cases where children have been sexually abused by clergymen.
Duran said he was sexually abused for three years by a Los Angeles Archdiocese priest beginning in 1983, when Duran was 11.
He and three other men allegedly sexually abused as boys by the same priest settled their lawsuits for $9.9 million against the archdiocese, Cardinal Roger Mahony and the now defrocked priest. Mahony was among the 115 cardinals in Rome who participated in the papal election this week.
Duran said he felt vindicated by the settlement. He and his attorney said authorities should investigate Mahony for his handling of child abuse complaints against the former priest, Michael Baker. The priest, who couldn't be reached for comment, served a prison sentence for molesting boys, Duran's attorneys said.
The 76-year-old leader, who served as the archbishop of Buenos Aires, is the first pope to take the name in honor of St. Francis of Assisi, revered among Catholics for his work with the poor.
The pontiff is a follower of the church's most social conservative wing. As a cardinal, he clashed with the government of Argentine President Cristina Fernandez de Kirchner over his opposition to gay marriage and free distribution of contraceptives.
He was runner-up in the 2005 papal conclave, behind then-Cardinal Joseph Ratzinger.
The new pope brings together the first and the developing worlds. Latin America is home to 480 million Catholics.
Francis' first public appearance as pope -- when he appealed for the crowds to pray for him before he gave a blessing -- suggested a "different pastoral style" in comparison with the more academic approach of Benedict, said Lombardi.
Francis is someone who has had "a day-to-day link with the population and ordinary people" during his many years at the head of a large diocese in Buenos Aires, he said.
He also sought to dampen concerns prompted by media reports that the new pope has only one lung.
Although Francis had part of one lung removed when he was a young man, the whole lung was not removed and the new pope is in good health, Lombardi said.
CNN iReporter Cesar Sotolongo in Lima, Peru, said the election of a Latin American pope, particularly from the Jesuit order, marked "a new chapter" for the Catholic Church.
Originally from Florida, Sotolongo also has his own advice for Francis: "The pope should shape the church with what he has been doing during his career (as an example)," he said. "Stay in contact with the people, communicate clearly, promote the unification of faith and ... represent the word of Jesus."
A Jesuit pope
Born in Buenos Aires to an Italian immigrant father, Francis is known for his simplicity.
Details given by Lombardi on Thursday of Francis' first hours as pope reinforce that impression -- one which may go down well with his global flock, many of whom live in poverty or are feeling the squeeze of austerity.
Francis stood, rather than sitting on a throne, to receive the oath of allegiance from his fellow cardinals after his election, and for his appearance on the balcony wore just a white cassock and a simple cross, eschewing gold or jewels, Lombardi said.
Also, on the ride back from the Sistine Chapel to the Santa Marta residence, he declined the papal car that had been prepared for him and instead took the bus with other cardinals, Lombardi said.
And Francis thanked the other cardinals at dinner, joking, "May God forgive you for what you have done," Lombardi said.
Francis will remove the seals from the official papal apartments Thursday but will not move in until renovations are complete, he added. The new pontiff will live in a suite at the Santa Marta residence until the papal apartments are ready.
In Buenos Aires, Francis chose to live in an apartment rather than the archbishop's palace, passed on a chauffeured limousine, took the bus to work and cooked his own meals.
He was ordained by the Jesuits in 1969. He became co-archbishop of Buenos Aires in 1997 and sole archbishop of that city one year later.
He was made a cardinal in 2001 and served as president of the Argentine bishops conference from 2005 to 2011.
As a Jesuit, Francis is a member of the Society of Jesus, one of the biggest and most important orders in the church.
Jesuits are recognized for their exceptional educational institutions and focus on social justice.
"Jesuits are characterized by their service to the church ... but trying to avoid positions of power," said Lombardi, the Vatican spokesman, who is also a Jesuit. "I am absolutely convinced that we have a pope who wants to serve.
"His election was the election of a rejection of power."
'Most stunning' choice of name
His selection of the name of Pope Francis is "the most stunning" choice and "precedent shattering," CNN Vatican analyst John Allen said. "The new pope is sending a signal that this will not be business as usual."
The name symbolizes "poverty, humility, simplicity and rebuilding the Catholic Church," Allen said.
Miguel Diaz, a former U.S. ambassador to the Holy See, agreed, calling the new pontiff's choice of names "very significant."
"Francis of Assisi is the saint who opted for the little ones in God's kingdom," he said. "This man represents a change and could potentially be a great gift for leadership, servant leadership, for all of us within the church and society."
It is something the Catholic Church says it desperately needs.
"If you look back over the past years -- the crisis of abuse, the scandals here at the Vatican, financial mismanagement, questions about the leaks and everything -- when you step back from it all, every crisis we faced ultimately is a crisis of holiness that we've missed the calling," said the Rev. Thomas Rosica, the Vatican's deputy spokesman.
"We've moved far away from what we're supposed to be."
Word of the election of Pope Francis, who was not considered a frontrunner among analysts, quickly spread around the globe, with everyone from U.N. Secretary-General Ban Ki-moon to U.S. President Barack Obama offering congratulations.
"As the first pope from the Americas, his selection also speaks to the strength and vitality of a region that is increasingly shaping our world, and alongside millions of Hispanic Americans, those of us in the United States share the joy of this historic day," Obama said.
Ban said the new pope shares common goals with the United Nations, from the promotion of peace to social justice. "We also share the conviction that we can only resolve the interconnected challenges of today's world through dialogue," he said.
There is likely to be no shortage of invitations for Pope Francis to travel to the four corners of the globe in the pursuit of such goals.
Syria's Patriarch Gregory III Laham of Antioch, who heads the Melkite Greek Catholic Church, on Thursday invited Francis to visit Syria, Jerusalem and Lebanon for peace and reconciliation, according to Syria's official news agency.
Israeli President Shimon Peres and Palestinian Authority President Mahmoud Abbas also urged him to visit the Middle East.
"He'll be a welcome guest in the Holy Land, as a man of inspiration that can add to the attempt to bring peace in a stormy area," said Peres.
Nowhere was the reaction to Francis' selection as pope more heartfelt than in Latin America.
"I am truly still very surprised ... not just that a Latino pope came out, but that he is an Argentinian from Buenos Aires," the Rev. Eduardo Mangiarotti, an Argentine priest, told CNN en Español.
It's a "huge event" not only for the church in Latin America but worldwide, he said.
Cardinal Theodore McCarrick, archbishop emeritus of Washington, greeted the selection with "extraordinary joy."
"I have been hoping that we would move into the Southern Hemisphere, and especially I think many of us had hoped ... we would have a pope who would come from Latin America," he said.
"One-half of the Catholics in the world are from Latin America, so this is a way the cardinals have very graciously acknowledged that."
Filipino priest and CNN iReporter Joel Camaya was among the tens of thousands who witnessed history Wednesday night in St. Peter's Square, as Francis emerged on the balcony.
"The multitude, from all parts of the world, were ecstatic to be in the square for this beautiful occasion," he said. "This was one event that left me teary-eyed and thanking God for making me a Catholic."
CNN's Richard Allen Greene and Hada Messia reported from Vatican City, Laura Smith-Spark wrote and reported in London, and Chelsea J. Carter and Ed Payne wrote and reported from Atlanta. CNN's Rosalina Nieves, Sarah Brown, Saad Abedine, Sara Sidner, Dana Ford, Catherine E. Shoichet, Mariano Castillo and Greg Botelho contributed to this report.