
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 $xoy$, given circle $C_{1}$: $(x+3)^{2}+(y-1)^{2}=4$ and circle $C_{2}$: $(x-4)^{2}+(y-5)^{2}=4$  
$(1)$ If line $l$ passes through point $A(4,0)$ and the chord intercepted by circle $C_{1}$ has a length of $2\sqrt{3}$, find the equation of line $l$.  
$(2)$ Let $P$ be a point in the plane such that there exist infinitely many pairs of perpendicular lines $l_{1}$ and $l_{2}$ passing through $P$, which intersect circles $C_{1}$ and $C_{2}$ respectively, and the lengths of the chords intercepted by $C_{1}$ and $C_{2}$ are equal. Find the coordinates of all such points $P$.

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