
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 a positive integer $m$ (where $m \geq 3$), consider an arithmetic sequence $\left\{a_{n}\right\}$ with positive terms and a geometric sequence $\left\{b_{n}\right\}$ with positive terms such that the first term of $\left\{a_{n}\right\}$ is equal to the common ratio of $\left\{b_{n}\right\}$, the first term of $\left\{b_{n}\right\}$ is equal to the common difference of $\left\{a_{n}\right\}$, and $a_{m}=b_{m}$. Find the minimum value of $a_{m}$ and determine the ratio of $a_{1}$ to $b_{1}$ when $a_{m}$ takes the minimum value.

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