
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 \(m (m \geq 3)\), let \(\{a_n\}\) be an increasing arithmetic sequence and \(\{b_n\}\) be an increasing geometric sequence such that the first term of \(\{a_n\}\) equals the common ratio of \(\{b_n\}\), and the first term of \(\{b_n\}\) equals the common difference of \(\{a_n\}\), and \(a_m = b_m\). Determine the minimum value of \(a_m\) and find the ratio of \(a_1\) to \(b_1\) when \(a_m\) takes this minimum value.

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