
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
A soccer team has $22$ available players. A fixed set of $11$ players starts the game, while the other $11$ are available as substitutes. During the game, the coach may make as many as $4$ substitutions, where any one of the $11$ players in the game is replaced by one of the substitutes. A player who has been substituted out cannot reenter the game, and each substitute can only participate once. No two substitutions can happen simultaneously. Let $n$ be the number of ways the coach can make substitutions during the game (including the option of making no substitutions). Find the remainder when $n$ is divided by $1000$.

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