
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 that the sequence $\{a_n\}$ is an arithmetic sequence, and its sum of the first $n$ terms is $S_n$, $\{b_n\}$ is a geometric sequence, and $a_1=b_1=2$, $a_4+b_4=27$, $S_4-b_4=10$.

$(1)$ Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$;

$(2)$ Does there exist $m(m\in\mathbb{N}^*)$ such that $\frac{4}{m}a_{7-m}$, $\frac{4}{m}a_{7}$, $\frac{4}{m}a_{7+4m}$ are three consecutive terms in the sequence $\{b_n^2\}$? Explain your reasoning.

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