
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: $x^2 + y^2 - 2x - 4y + m = 0$, where $m < 5$:
1. If circle C intersects line $l: x + 2y - 4 = 0$ at points M and N, and the distance $|MN| = \frac{4\sqrt{5}}{5}$, find the value of m;
2. Under the condition of (1), determine whether there exists a line $l: x - 2y + c = 0$ such that there are four points on the circle at a distance of $\frac{\sqrt{5}}{5}$ from line l. If it exists, find the range of c, and if not, explain why.

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