
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 sequence $\{a_n\}$ with the sum of the first $n$ terms being $S_n$, and $a_n=2\sqrt{S_n}-1$. Let $c$ be a real number. For any three distinct positive integers $m$, $k$, $n$ that form an arithmetic sequence, the inequality $S_m+S_n > cS_k$ always holds. Then, the range of values for the real number $c$ is \_\_\_\_\_\_.

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