
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 that $S_{n}$ is the sum of the first $n$ terms of the arithmetic sequence $\{a_{n}\}(n\in\mathbb{N}^{*})$, and $S_{6} > S_{7} > S_{5}$, there are the following five propositions:  
$(1) d < 0$;  
$(2) S_{11} > 0$;  
$(3) S_{12} < 0$;  
$(4)$ The largest term in the sequence $\{S_{n}\}$ is $S_{11}$;  
$(5) |a_{6}| > |a_{7}|.$  
Among these, the correct propositions are __________ (Write down the numbers of all propositions you think are correct).

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