WASHINGTON — After House Republicans began their closed-door meeting on Tuesday with an impromptu rendition of “Amazing Grace,” Speaker John A. Boehner stood up with what he hoped would be an equally inspiring message.

The Senate was trying to steamroll the House with a new plan to reopen the government and lift the debt ceiling, leaving his team with only two options: “We can wait on the Senate for four days and accept whatever they give us,” he told them. “Or we can go on offense and force their hand.”

It was yet another moment of decision for Mr. Boehner, who finally finds himself at the crossroads he has been marching toward for weeks: an imminent financial default on the one hand, and on the other an unyielding conservative rank and file that persists with the futile effort to take down President Obama’s health care law even if they also take down the speaker in the process.

While his colleagues sang about how what once was lost had now been found, Mr. Boehner did not tell them a more dispiriting truth: With less than 48 hours left before the nation is set to exhaust its authority to borrow money, he and his lieutenants were running out of ideas — a fact made starkly evident by the mad and fruitless scramble on Tuesday to come up with a measure that could win enough support from his members. Around 7 p.m., he sent the House home and canceled all votes for the day.