
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 the sets \( M=\{x \mid x=3n, n=1,2,3,4\} \) and \( P=\{x \mid x=3^k, k=1,2,3\} \), find the number of sets \( S \) that satisfy the conditions \( (M \cap P) \subseteq S \subseteq (M \cup P) \).

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