
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 the equation of circle C is $x^2 - y^2 - 2x - 4y + m = 0$

(1) If the radius of circle C is 2, find the value of $m$.

(2) If circle C intersects line $l: x + 2y - 4 = 0$ at points M and N, and $|MN| = \frac{4\sqrt{5}}{5}$, find the value of $m$.

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