
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
Let \( M \) be the set of all permutations \( (a_1, a_2, \ldots, a_{10}) \) of the set \(\{1, 2, \ldots, 10\}\) that satisfy the condition \( 3k \mid (a_1 + a_2 + \ldots + a_{3k}) \) for \( k = 1, 2, 3 \). Find the number of elements in the set \( M \).

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