
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
The MathMatters competition consists of 10 players  $P_1$ ,  $P_2$ ,  $\dots$ ,  $P_{10}$  competing in a ladder-style tournament.  Player  $P_{10}$  plays a game with  $P_9$ : the loser is ranked 10th, while the winner plays  $P_8$ .  The loser of that game is ranked 9th, while the winner plays  $P_7$ .  They keep repeating this process until someone plays  $P_1$ : the loser of that final game is ranked 2nd, while the winner is ranked 1st.  How many different rankings of the players are possible?

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