
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 about $x$ and $y$: $C: x^2 + y^2 - 2x - 4y + m = 0$.
1. For which value of $m$ does equation $C$ represent a circle?
2. If circle $C$ intersects line $l: x + 2y - 4 = 0$ at points $M$ and $N$, and $MN = \frac {4}{\sqrt {5}}$, find the value of $m$.

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