
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
Fix an integer  $n \geq 2$  and let  $a_1, \ldots, a_n$  be integers, where  $a_1 = 1$ . Let  $$  f(x) = \sum_{m=1}^n a_mm^x.  $$    Suppose that  $f(x) = 0$  for some  $K$  consecutive positive integer values of  $x$ . In terms of  $n$ , determine the maximum possible value of  $K$ .

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