
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
(1) In triangle $ΔABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c=2$, $C=\frac{\pi}{3}$, and the area of $ΔABC$ is $\sqrt{3}$, then the perimeter of $ΔABC$ is __________.

(2) Given the line $l: \frac{x}{a} + \frac{y}{b} = 1 (a > 0, b > 0)$ passes through the point $(3, 2)$, then the minimum value of $a+b$ is __________.

(3) A person randomly throws a sesame seed into a triangular area with sides of lengths $5$, $12$, and $13$. Assuming the sesame seed can land on any point within the area with equal probability, the probability that it lands in a place where the distance to all three vertices is greater than $2$ is ____.

(4) Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\} (n \in N_+)$, and it is known that $S_6 > S_7 > S_5$, then the correct statement(s) among the following four is/are __________.

- $① d < 0$;
- $② S_{11} > 0$;
- $③ S_{12} < 0$;
- $④$ The maximum term in the sequence $\{S_n\}$ is $S_{11}$.

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