
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
Given \( M = \{1, 2, \cdots, 10\} \). For a partition \( A_1, A_2, A_3 \) (each \( A_i \) being non-empty) of \( M \), it is called a "good" partition if there exists a permutation \( A_{i1}, A_{i2}, A_{i3} \) of \( A_1, A_2, A_3 \) such that 
$$
\max A_{ik} > \min A_{i, k+1} \quad \left(k = 1, 2, 3, \, A_{i4} = A_{i1}\right).
$$
Find the number of "good" partitions of \( M \).

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