
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
Given that circle $C$ has its center at $C(2,4)$ and is tangent to the line $3x-4y=0$. Line $l$ passes through the origin and intersects circle $C$ at points $A$ and $B$. $P$ is the midpoint of $AB$.

(1) Find the equation of circle $C$;

(2) If $\triangle ABC$ is a right triangle, find the equation of line $l$;

(3) Determine if there exists a line ${l}'$ passing through the point $(0,-1)$ such that it intersects line $l$ at point $Q$ and satisfies $|OP| \cdot |OQ| = 4$. If it exists, find the equation of line ${l}'$; otherwise, explain the reason.

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