
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 $A=\{a_1, a_2, \cdots, a_n\}$ be a set of numbers and let $P(A)$ denote the arithmetic mean of all elements in $A$ such that $P(A)=\frac{a_1 + a_2 + \cdots + a_n}{n}$. If $B$ is a non-empty subset of $A$ and $P(B)=P(A)$, then $B$ is called a "balanced subset" of $A$. Find the number of "balanced subsets" of the set $M=\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$.

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