
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 the sum of the first n terms of an arithmetic sequence {a_n} is denoted by S_n, for all n in the set of natural numbers, the inequality (n+1)S_n < nS_{n+1} holds true, and the ratio of a_8 to a_7 is less than -1, find the maximum value of n for which S_n < 0.

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