
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
The real numbers  $a_{1},a_{2},\ldots ,a_{n}$  where  $n\ge 3$  are such that  $\sum_{i=1}^{n}a_{i}=0$  and  $2a_{k}\le\ a_{k-1}+a_{k+1}$  for all  $k=2,3,\ldots ,n-1$ . Find the least  $f(n)$  such that, for all  $k\in\left\{1,2,\ldots ,n\right\}$ , we have  $|a_{k}|\le f(n)\max\left\{|a_{1}|,|a_{n}|\right\}$ .

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