
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 such that the product of the distances to the line $l_{1}: x=-1$ and the line $l_{2}: y=1$ equals a constant $k^{2}$ ($k > 0$). Among the following four statements:

$①$ The curve $C$ passes through the point $(-1,1)$;

$②$ The curve $C$ is centrally symmetric about the point $(-1,1)$;

$③$ If point $P$ is on the curve $C$, and points $A$, $B$ are on the lines $l_{1}$, $l_{2}$ respectively, then $|PA|+|PB|$ is not less than $2k$;

$④$ Let $P_{0}$ be any point on the curve $C$, and let points $P_{1}$, $P_{2}$, $P_{3}$ be symmetric to $P_{0}$ with respect to the line $l_{1}: x=-1$, the point $(-1,1)$, and the line $f(x)$ respectively, then the area of the quadrilateral $P_{0}P_{1}P_{2}P_{3}$ is a constant $4k^{2}$;

The correct statements are \_\_\_\_\_\_.

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