
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 the non-empty subset family \(\{A_{1}, A_{2}, A_{3}\}\) be a partition of the set \(M = \{1, 2, \ldots, 10\}\), meaning the non-empty sets \(A_{1}, A_{2}, A_{3}\) are pairwise disjoint and their union is \(M\). If there exists a permutation \(i_1, i_2, i_3\) of \(1, 2, 3\) such that
\[ \max A_{i_k} > \min A_{i_{k+1}}, \quad k = 1, 2, 3, \]
where \(i_4 = i_1\), then \(\{A_{1}, A_{2}, A_{3}\}\) is called a good partition of \(M\). Find the number of all good partitions of \(M\).

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