
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 $xOy$, there are two circles $C_1: x^2 + y^2 = 16$ and $C_2: (x - 7)^2 + (y - 4)^2 = 4$,
(1) Find the equation of the tangent line to the circle $C_1$ that passes through the point $(4, 6)$;
(2) Let $P$ be a point on the coordinate plane, such that there exist infinitely many pairs of perpendicular lines $l_1$ and $l_2$ passing through $P$. These lines intersect with circle $C_1$ and $C_2$ respectively, and the length of the chord intercepted by $l_1$ on circle $C_1$ is twice the length of the chord intercepted by $l_2$ on circle $C_2$. Find all coordinates of point $P$ that satisfy the condition.

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