
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 a nonempty set \( M \) satisfying
\[ 
M \subseteq \{0,1, \cdots, n\} \quad \text{(where \( n \geq 2 \), \( n \) is a positive integer)}.
\]

If there exists a nonnegative integer \( k \) (\( k \leq n \)) such that whenever \( a \in M \), \( 2k - a \in M \) holds, then the set \( M \) is said to have the property \( P \). Let \( f(n) \) represent the number of sets \( M \) that have the property \( P \). Find the value of \( f(9) - f(8) \).

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