
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 geometric sequence $\{a_n\}$ with a common ratio not equal to 1, where $a_1=1$, $a_2=a$, and $a_{n+1}=k(a_n+a_{n+2})$ holds for any positive integer $n$. If for any three consecutive terms $a_m$, $a_{m+1}$, $a_{m+2}$, they can be arranged in some order to form an arithmetic sequence, then $k= \boxed{-\frac{2}{5}}$.

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