
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
Let $\{a_n\}$ be an arithmetic sequence, and $S_n$ be the sum of its first $n$ terms, such that $S_5 < S_6$, $S_6=S_7 > S_8$. Select the correct statements from the list below:  
(1) $d<0$  
(2) $a_7=0$  
(3) $S_9 > S_5$  
(4) Both $S_6$ and $S_7$ are the maximum values of $S_n$.

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