
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Two bored millionaires, Bilion and Trilion, decide to play a game. They each have a sufficient supply of  $\$  1, \ $ 2,\$  5 $, and $ \ $ 10$  bills. Starting with Bilion, they take turns putting one of the bills they have into a pile. The game ends when the bills in the pile total exactly  $\$ 1{,}000{,}000 $, and whoever makes the last move wins the $ \ $1{,}000{,}000$  in the pile (if the pile is worth more than  $\$ 1{,}000{,}000$ after a move, then the person who made the last move loses instead, and the other person wins the amount of cash in the pile). Assuming optimal play, how many dollars will the winning player gain?

*Proposed by Yannick Yao*

Present the answer in LaTex format: \boxed{Your answer}
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