
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 $a$ ($a$ and $b$ are both positive integers).
(I) If $a_{1}=b_{1}$, $a_{2}=b_{2}$, find the general terms of the sequences $\{a_{n}\}$ and $\{b_{n}\}$.
(II) For the sequences $\{a_{n}\}$ and $\{b_{n}\}$ from part (I), insert $a_{k}$ number of '2' between $b_{k}$ and $b_{k+1}$ for any $k∈N^{*}$ to form a new sequence $\{c_{n}\}$. Find all positive integer values of $m$ that satisfy the equation $c_{1}+c_{2}+…+c_{m}=2c_{m+1}$.

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