Geoff Bennett:

Two cease-fires are on shaky ground tonight. Iran says the U.S. has violated its truce after the U.S. carried out what it called self-defense strikes against Iranian missile launch sites and boats. And Israel is escalating its offensive against Hezbollah in Lebanon.

Amna Nawaz:

Still, U.S. officials insist both cease-fires remain intact. And they await a response from Iran on the latest outlines of a deal to end the war.

But, at least publicly, Iran is digging in, accusing the U.S. of acting in bad faith. And Iran's navy claims that dozens of tankers have passed through the Strait of Hormuz since yesterday, only with their permission.

William Brangham starts us off.

William Brangham:

Across the airwaves in Iran today, state broadcasters promised consequences for recent U.S. strikes that came in the middle of peace negotiations.

Woman (through interpreter):

The Foreign Ministry, in a statement condemning the cease-fire violation by the U.S. terrorist military, emphasized that the Islamic Republic of Iran will leave no act of aggression unanswered.

William Brangham:

And, on the streets of Tehran, continued public displays of defiance, like this huge banner depicting President Trump's mouth stitched shut with a ribbon shaped like the Strait of Hormuz, claiming the U.S. had reached -- quote -- "the breaking point."

The U.S. military said it carried out so-called self-defense strikes last night against Iranian missile launch sites and boats that were attempting to lay mines in the Strait of Hormuz. The U.S. said they were carried out with restraint in light of the weeks-long cease-fire.

But some Iranians viewed it as more evidence that the U.S. could not be trusted.

Reza Rahimi, Tehran, Iran, Resident (through interpreter):

The politicians should not pay any attention to America and their agreements at all, because they really go back on all their agreements.

William Brangham:

U.S. officials insist the cease-fire is holding. Secretary of State Marco Rubio told reporters the strikes won't set back the peace talks. But he said the details could take some time.

Marco Rubio:

Like anything with something like this, it's going to take a couple days to settle on even down to the disagreements over a word, a sentence. It's either going to be a good deal or there isn't going to be one.

William Brangham:

Both Iranian and U.S. officials have cited progress towards an initial deal that would lead to further negotiations, including on the fate of Iran's nuclear program.

In what amounted to a potential softening of his position, President Trump wrote on social media last night that Iran's highly enriched uranium would be -- quote -- "destroyed in place or at another acceptable location, with the Atomic Energy Commission or its equivalent being witness."

That echoes what Iranian officials have told the "News Hour," that they are willing to export their highly enriched uranium to the U.N.'s nuclear watchdog, the IAEA, but not to the U.S.

Still, there has been no public confirmation that Iran is willing to restrict its nuclear program, which it has always claimed is for civilian use only. As part of the latest proposal, there would be an end to all fighting, including in Lebanon.

But Israeli Prime Minister Benjamin Netanyahu has ordered more frequent and heavier strikes against Iran-backed Hezbollah militants across Lebanon, despite the U.S.-brokered cease-fire there that remains in name only. Massive strikes overnight leveled entire blocks in Southern and Eastern Lebanon, killing at least a dozen people.

Netanyahu huddled with his top military officials today and later announced that Israeli troops would be pushing further north.

Benjamin Netanyahu, Israeli Prime Minister (through interpreter):

Under my direction, the IDF is operating with large forces on the ground and is seizing controlled areas.

William Brangham:

Israeli military officials said their operations will push past the so-called yellow line of demarcation near Lebanon's southern border. Israel has said it will not withdraw until Hezbollah is no longer close enough to pose a threat to its northern villages.

For the "PBS News Hour," I'm William Brangham.