
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
<span style="color:darkred"> Let  $m$  and  $n$  be two nonzero natural numbers. Determine the minimum number of distinct complex roots of the polynomial  $\prod_{k=1}^m\, (f+k)$  , when  $f$  covers the set of  $n^{\text{th}}$  - degree polynomials with complex coefficients.
</span>

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