AMNA NAWAZ: Welcome to the "News Hour."

The war with Iran has entered its second month, and President Trump says the onus is on Tehran to quickly agree to end the war or face the widespread destruction of its energy resources, including its power plants and its oil hubs.

But Iran remains publicly defiant, calling U.S.

demands unrealistic and continuing its attacks on Israel and the region.

Nick Schifrin begins our coverage.

NICK SCHIFRIN: In Northern Israel today, the country's largest oil refinery engulfed in smoke.

Iran is trying to punish Israel and the region with attacks on energy facilities and a stranglehold on the Strait of Hormuz, which White House spokesperson Karoline Leavitt said today Iran was beginning to release.

KAROLINE LEAVITT, White House Press Secretary: Twenty additional tankers which we expect to see over the coming days are a result of the direct and indirect talks that are taking place between the United States and Iran.

NICK SCHIFRIN: President Trump told The Financial Times the Strait of Hormuz agreement was personally authorized by Iran's Parliament speaker, Mohammad Bagher Ghalibaf.

He's long been a hard-liner who this weekend warned he would -- quote -- "set American troops on fire."

But today President Trump said Iran was led by a -- quote -- "new and more reasonable regime."

He also threatened to escalate, writing: "If the Hormuz Strait is not immediately open for business, we will conclude our lovely stay in Iran by blowing up and completely obliterating all of their-electric generating plants, oil wells and Kharg Island."

Kharg Island is about the size of New York's JFK Airport and exports 90 percent of Iran's oil.

Trump told the F.T.

that he would -- quote -- "prefer" to take Iran's oil by seizing the island.

"Maybe we take Kharg Island.

Maybe we don't.

We have a lot of options.

It would also mean we had to be there for a while."

Even for President Trump, a threat to possibly indefinitely occupy Middle Eastern territory is a remarkable statement, as a leader who has for one decade vowed to avoid endless wars.

But the president dismissed his detractors: "My favorite thing is to take the oil in Iran, but some stupid people back in the USA say, why are you doing that?

But they're stupid people."

Over the last week, the U.S.

has demanded Iran dismantle its nuclear facilities and hand over its highly enriched uranium, cap its ballistic missile inventory and range, and give up its support for proxies, as Secretary of State Marco Rubio said today.

MARCO RUBIO, U.S.

Secretary of State: So, they have to abandon all these weapon programs and all their nuclear ambitions.

And if they do that, then Iran's future can be bright.

NICK SCHIFRIN: But, today, the U.S.

demands were rejected by Iran's Foreign Ministry spokesman.

ESMAEIL BAGHAEI, Iranian Foreign Ministry Spokesman (through translator): The U.S.

demands have been very excessive, unrealistic and illogical.

KAROLINE LEAVITT: What is said publicly is, of course, much different than what's being communicated to us privately.

NICK SCHIFRIN: Today, Leavitt said Iran was embracing diplomacy.

KAROLINE LEAVITT: When the president says more reasonable, again, these folks are appearing more reasonable behind the scenes privately in these conversations than perhaps some of the previous leaders who are now no longer on planet Earth because they lied to the United States.

NICK SCHIFRIN: But Iran also accuses the U.S.

of using diplomacy as cover.

Twice during Trump's second term, the U.S.

has attacked Iran during high-level diplomatic talks.

DONALD TRUMP, President of the United States: I would only say that we're doing extremely well in that negotiation.

But you never know with Iran, because we negotiate with them, and then we always have to blow them up.

NICK SCHIFRIN: The U.S.

is amassing more forces, including thousands of Marines with sailors who can launch amphibious assaults.

They could conduct raids into Iranian territory, but they could also help defend tankers passing through the strait.

In Iran, today, the White House said it had struck 11,000 targets out of what President Trump says are 16,000 targets.

And Israel continued to target what it says are Tehran weapons production sites.

Iran says among the targets have been three universities, as well as now hollowed-out high-rise apartments.

At least one strike hit a busy street in downtown Tehran, and the city has faced power cuts.

Israel is also expanding its invasion into Southern Lebanon, expanding what Prime Minister Benjamin Netanyahu this weekend called a security strip.

BENJAMIN NETANYAHU, Israeli Prime Minister (through translator): I have now instructed to further expand the existing security strip in order to finally thwart the threat of invasion and to keep the anti-tank missile fire away from our borders.

NICK SCHIFRIN: That expansion has come with a cost.

Today, Israel buried 19-year-old Sergeant Liran Ben Zion, an American Israeli and the sixth Israeli soldier killed in Lebanon in the last month, when Hezbollah entered the war and has since fired at least 850 drones and missiles into Israel.

The U.N.

's peacekeeping mission in Southern Lebanon said today three peacekeepers had been killed in less than 24 hours.

And in Southern Lebanon this weekend, an Israeli strike killed three journalists and multiple paramedics.