The US player would lose the game in this instance. If you draw all scoring cards and do not have enough phases left to play them, you lose.
This is one of the risks of using "Ask Not" late in the turn. A small risk, but potentially disastrous as your example illustrates.
From the collected rulings:
"Ask Not" can lead to the US player holding a Scoring card (if only Scoring cards are drawn). This is allowed (an exception to 10.1.5) and results in a US loss at the end of the turn, as usual for holding scoring cards.