
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
Among all the permutations $(a_{1}, a_{2}, \ldots, a_{n})$ of the set $\{1,2, \ldots, n\}$, $(n \geq 1$ integer), consider those that satisfy that $2(a_{1}+a_{2}+\cdots+a_{m})$ is divisible by $m$ for each $m=1,2, \ldots, n$. Calculate the total number of these permutations.

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