
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 a circle $C: x^2 + y^2 - 2x + 4my + 4m^2 = 0$, a circle $C_1: x^2 + y^2 = 25$, and a line $l: 3x - 4y - 15 = 0$:
1. Find the chord length of $C_1$ cut by line $l$.
2. Find the value of $m$ for which the common chord of circles $C$ and $C_1$ is parallel to line $l$.
3. Determine if there is a value of $m$ such that the midpoint of chord $AB$ cut by line $l$ on circle $C$ is at a distance equal to half the length of chord $AB$ from point $P(2, 0)$. If such $m$ exists, find the equation of circle $C$; if not, provide an explanation.

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