WASHINGTON (AP) — The U.S. military on Tuesday pulled off its biggest day of evacuation flights from Afghanistan since the operation began. But deadly violence that has blocked many desperate evacuees from entering Kabul's airport persisted, and the Taliban signaled they might soon seek to shut down the airlifts.

Amid the tense operation to get people out of the country, CIA Director William Burns secretly swooped into Kabul on Monday to meet with the Taliban's top political leader, a U.S. official told The Associated Press.

About 21,600 people were flown safely out of Taliban-held Afghanistan in the 24-hour period that ended early Tuesday, the White House said. That compares with about 16,000 the previous day.

Thirty-seven U.S. military flights — 32 C-17s and 5 C-130s — carried about 12,700 evacuees. An additional 8,900 people flew out aboard 57 flights by U.S. allies.

Pentagon chief spokesman John Kirby said Monday the faster pace of evacuation was partly due to coordination with Taliban commanders on getting evacuees into the airport.

“Thus far, and going forward, it does require constant coordination and deconfliction with the Taliban,” Kirby said. “What we've seen is, this deconfliction has worked well in terms of allowing access and flow as well as reducing the overall size of the crowds just outside the airport.”

CIA Director Burns and Abdul Ghani Baradar, the Taliban's top political leader, secretly met in Kabul on Monday as the evacuations continued. The Washington Post first reported Burns' meeting. The U.S. official later confirmed the meeting for the AP.

The senior U.S. military commander at the Kabul airport, Rear Adm. Peter Vasely, has been communicating daily with Taliban commanders in an effort to facilitate the evacuation, but the last known contact between the military and Baradar was when Gen. Mark Milley, chairman of the Joint Chiefs of Staff, flew to Doha, Qatar, to meet with him and other Taliban officials last December. Milley tried to persuade the Taliban to reduce their attacks against Afghan forces, ultimately to no avail.

With access to the airport still difficult Monday, the U.S. military went beyond the perimeter to carry out another helicopter retrieval of Americans. U.S. officials said a military helicopter picked up 16 American citizens Monday and brought them to the airfield for evacuation. This was at least the second such rescue mission beyond the airport; Kirby said that last Thursday, three Army helicopters picked up 169 Americans near a hotel just beyond the airport gate and flew them onto the airfield.

President Joe Biden's national security adviser, Jake Sullivan, said at the White House on Monday that talks with the Taliban were continuing as the administration looks for additional ways to safely move more Americans and others into the Kabul airport by an end-of-August deadline.

He said ultimately it will be Biden's decision alone whether to continue military-led evacuation operations beyond Aug. 31. That's the date Biden has set for completing the withdrawal of troops.

California Democrat Rep. Adam Schiff, chairman of the House intelligence committee, told reporters after a committee briefing Monday on the Afghanistan withdrawal that “it was hard for me to imagine” wrapping up the airlifts by the end of the month. He also said it was clear there had been "any number of warnings” to the administration “of a very rapid takeover” by the Taliban.

After more than a week of evacuations plagued by major obstacles, including Taliban forces and crushing crowds that are making approaching the airport difficult and dangerous, the number of people flown out met — and exceeded — U.S. projections for the first time.

Army Gen. Stephen Lyons, head of U.S. Transportation Command, which manages the military aircraft that are executing the Kabul airlift, told a Pentagon news conference that more than 200 planes are involved, including aerial refueling planes, and that arriving planes are spending less than an hour on the tarmac at Kabul before loading and taking off. He said the nonstop mission is taking a toll on aircrews.

“They're tired,” Lyons said of the crews.

On a positive note, Lyons said that in addition to the widely reported case of an Afghan woman giving birth aboard a U.S. evacuation aircraft, two other babies have been born in similar circumstances. He did not provide details.

The Pentagon said it has added a fourth U.S. military base, in New Jersey, to three others — in Virginia, Texas and Wisconsin — that are prepared to temporarily house arriving Afghans. Maj. Gen. Hank Williams, the Joint Staff deputy director for regional operations, told reporters there are now about 1,200 Afghans at those military bases. The four bases combined are capable of housing up to 25,000 evacuees, Kirby said.

Afghan evacuees continued to arrive at Dulles International Airport outside of Washington. Exhaustion clouded the faces of many of the adults. A journalist asked one man how it felt to be in the U.S. “We are safe,” the man answered.

An older woman sank with relief into an offered wheelchair, and a little girl carried by an older boy shaded her eyes to look curiously around. The scramble to evacuate left many arrivals carrying only a bookbag or purse, or a plastic shopping bag of belongings. Some arrived for their new lives entirely empty-handed.

Biden said Sunday he would not rule out extending the evacuation beyond Aug. 31. British Prime Minister Boris Johnson, who will meet with Biden virtually on Tuesday in a G-7 leaders' summit on the chaotic withdrawal, is expected to press Biden for an extension to get out the maximum number of foreigners and Afghan allies possible.

Lawmakers, veterans organizations and refugee advocates in the U.S. also are urging Biden to keep up the U.S. military's evacuation out of the Kabul airport as long as it takes to airlift not just Americans, but Afghan allies and other Afghans most at risk from the Taliban.

But Taliban spokesman Suhail Shaheen, in an interview with Sky News, said that Aug. 31 is a “red line” the U.S. must not cross and that extending the American presence would “provoke a reaction.”

Monday's warning signaled the Taliban could insist on shutting down the airlifts out of the Kabul airport in just over a week. Lawmakers, refugee groups, veterans' organizations and U.S. allies have said ending the evacuation then could strand countless Afghans and foreigners still hoping for flights out.

Since Aug. 14, the U.S. has evacuated and facilitated the evacuation of more than 58,000 people.

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Associated Press writers Nomaan Merchant, Darlene Superville, Aamer Madhani, Lolita C. Baldor, Hope Yen, Alexandra Jaffe, James LaPorta, Jonathan Lemire, Matthew Lee and Dan Huff contributed to this report.