
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
Suppose $x\_1$, $x\_2$, ..., $x\_10$ is a permutation of $1$, $2$, ..., $10$. The number of different permutations that satisfy $x\_m + m \leq x\_n + n$ for any positive integer $m$, $n$ and $1 \leqslant m < n \leqslant 10$ is \_\_\_\_\_\_.

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