
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 the sum of the first $n$ terms denoted as $S_{n}$ for $n=1,2,3,\cdots$, and $a_{2}=3$, $S_{5}=25$.<br/>$(1)$ Find the arithmetic mean of $a_{2}$ and $a_{4}$.<br/>$(2)$ For a geometric sequence $\{b_{n}\}$ with the first term as $1$ and common ratio $q$, choose one of the following three conditions such that every term of $\{b_{n}\}$ is also a term of $\{a_{n}\}$. If ${b}_{k}={a}_{m}\;(k,m∈{N}^{*})$, find $m$ ($m$ expressed in terms of $k$).<br/>Condition 1: $q=-1$; Condition 2: $q=2$; Condition 3: $q=3$.

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