
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 an arithmetic sequence \\(\{a_n\}\\), it is known that \\(\dfrac{a_{11}}{a_{10}} + 1 < 0\\), and the sum of the first \\(n\\) terms of this sequence, \\(S_n\\), has a maximum value. Determine the maximum value of \\(n\\) for which \\(S_n > 0\\) holds.

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