
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
If \\(\{a_n\}\) is an arithmetic sequence, with the first term \\(a_1 > 0\), \\(a_{1007} + a_{1008} > 0\), and \\(a_{1007} \cdot a_{1008} < 0\), determine the largest positive integer \\(n\) for which the sum of the first \\(n\) terms \\(S_n > 0\).

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