
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
In the Cartesian coordinate system, given the circle $C_{1}$: $(x+4)^{2}+(y-2)^{2}=9$ and the circle $C_{2}$: $(x-5)^{2}+(y-6)^{2}=9$. Let $P$ be a point on the plane. If there exist infinitely many pairs of mutually perpendicular lines $l_{1}$ and $l_{2}$ passing through point $P$, intersecting circles $C_{1}$ and $C_{2}$ respectively, and the chord length intercepted by line $l_{1}$ on circle $C_{1}$ is equal to the chord length intercepted by line $l_{2}$ on circle $C_{2}$, then the coordinates of all points $P$ that satisfy this condition are ____.

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