
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

There are two sequences of real numbers, one arithmetic $\left(a_{n}\right)_{n \in \mathbb{N}}$ and the other geometric $\left(g_{n}\right)_{n \in \mathbb{N}}$ which is non-constant. It is given that $a_{1}=g_{1} \neq 0$, $a_{2}=g_{2}$, and $a_{10}=g_{3}$. Determine, with reasoning, if for each positive integer $p$, there exists a positive integer $m$ such that $g_{p}=a_{m}$.

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