
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
Let $f_{1}=\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ (with $n > 2$) be a sequence of integers. Perform the following operation on the sequence $f_{k}=\left(c_{1}, c_{2}, \cdots, c_{n}\right)$: $f_{k+1}=\left(c_{i_1}, c_{i_2}, c_{i_3}+1, \cdots, c_{i_{n}}+1\right)$, where $\left(c_{i_1}, c_{i_2}, \cdots, c_{i_n}\right)$ is any permutation of $\left(c_{1}, c_{2}, \cdots, c_{n}\right)$. Construct $f_{1}, f_{2}, \cdots$ in succession and find the necessary and sufficient conditions under which there exists a positive integer $k$ such that all $n$ numbers in $f_{k}$ are equal.

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