
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 $a$ and common difference $b$, and a geometric sequence $\{b_{n}\}$ with first term $b$ and common ratio $a$, where $a$ and $b$ are both positive integers, and $a_{1} < b_{1} < a_{2} < b_{2} < a_{3}$.

(I) Find the value of $a$;

(II) If there exist $m$, $n∈N^{*}$ such that $a_{m}+1=b_{n}$, find the value of $b$;

(III) For the sequences $\{a_{n}\}$ and $\{b_{n}\}$ that satisfy (II), let $c_{n}=\frac{a_{n}-8}{b_{n}}$, find the maximum term of the sequence $\{c_{n}\}$.

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