
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 $\{a_n\}$, with the sum of the first $n$ terms denoted as $S_n$, and $S_6 > S_7 > S_5$, the correct conclusion(s) among the following is/are: ______  
$(1) d < 0$;  $(2) S_{11} > 0$;  $(3) S_{12} < 0$;  $(4) S_{13} < 0$;  $(5) S_{9} > S_{3}$.

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