
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: $x^2 + y^2 - 2x - 4y + m = 0$.
1. Find the range of values for $m$.
2. If circle C intersects with the line $l: x + 2y - 4 = 0$ at points M and N, and $|MN| = \frac{4\sqrt{5}}{5}$, find the value of $m$.
3. If the circle from (1) intersects with the line $x + 2y - 4 = 0$ at points M and N, and $OM \perp ON$ (with O as the origin), find the value of $m$.

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