
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 curve C: $x^2 + y^2 - 2x - 4y + m = 0$, and let O be the coordinate origin:

1. For what value of $m$ does curve C represent a circle?
2. If curve C intersects the line $x + 2y - 3 = 0$ at points M and N, and $\overrightarrow{OM} \perp \overrightarrow{ON}$, find the value of $m$.

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