
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 first term $a_1=2$ and common ratio $q$ (where $q$ is a positive integer), such that $3a_3$ is the arithmetic mean of $8a_1$ and $a_5$. Additionally, the sequence $\{b_n\}$ satisfies the equation $2n^2 - (t + b_n)n + \frac{3}{2}b_n = 0$ $(t \in \mathbb{R}, n \in \mathbb{N}^*)$.

(1) Find the general term formula for the sequence $\{a_n\}$.
(2) Determine the value of the real number $t$ such that the sequence $\{b_n\}$ is an arithmetic sequence.
(3) When $\{b_n\}$ is an arithmetic sequence, insert $b_k$ number of '2's between $a_k$ and $a_{k+1}$ for each positive integer $k$ to form a new sequence $\{c_n\}$. Let $T_n$ be the sum of the first $n$ terms of $\{c_n\}$. Find all positive integers $m$ that satisfy $T_m = 2c_{m+1}$.

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