
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
If $\{a_n\}$ is an arithmetic sequence, and $m$, $n$, $p$ are distinct positive integers, then the correct conclusion is: $(m-n)a_p+(n-p)a_m+(p-m)a_n=0$. By analogy with the above property, correspondingly, if $\{b_n\}$ is a geometric sequence, and $m$, $n$, $p$ are distinct positive integers, then we have:

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