
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
The curve $C$ is the locus of points in the plane whose distances to two fixed points $F_1(-1,0)$ and $F_2(1,0)$ have a product equal to a constant $a^2$ ($a > 1$). Consider the following four statements:

$①$ The curve $C$ passes through the origin;

$②$ The curve $C$ is symmetric with respect to the origin;

$③$ If point $P$ is on the curve $C$, then the area $S$ of $\Delta F_1PF_2$ satisfies $S \leqslant \frac{1}{2}a^2$;

$④$ If point $P$ is on the curve $C$, then the perimeter $l$ of $\Delta F_1PF_2$ satisfies $l \geqslant 2a+2$.

The correct statements are __________.

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