
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 an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$, and the sum of the first $n$ terms is $S_n$. If $a_2$, $a_5$, and $a_{11}$ form a geometric sequence, and $a_{11} = 2(S_m - S_n) (m > n > 0, m, n \in \mathbb{N}^*)$, then the value of $m+n$ is ______.

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