
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 $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ ($n \in \mathbb{N}^*$), and it is given that $S_6 > S_7 > S_5$. Among the following four statements:

① $d<0$;   ② $S_{11} > 0$;    ③ $S_{12} < 0$;   ④ The maximum value of $n$ for which $S_n > 0$ is 13;

The correct statement(s) is/are __________.

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