
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 a circle $C: x^{2}+y^{2}-2x-4y+1=0$, there exist two points on it that are symmetric with respect to the line $l: x+my+1=0$. A tangent line to circle $C$ that passes through the point $M(m,m)$ touches the circle at point $P$. Find the values of $m$ and $|MP|$.

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