
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 first term $a$ and common difference $b$, and a geometric sequence $\{b_n\}$ with the first term $b$ and common ratio $a$, where $a$ and $b$ are positive integers greater than 1, and $a_1 < b_1$, $b_2 < a_3$. Find the value of $a$. If for any $n \in \mathbb{N}^*$, there exists an $m \in \mathbb{N}^*$ such that $b_n = a_m + 3$ holds, then find the expression for $a_n$.

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