
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 circle $C$: $x^{2}+y^{2}+2x-4y+1=0$, $O$ is the coordinate origin, and moving point $P$ is outside circle $C$. Draw a tangent line $l$ through point $P$, and let $M$ be the tangent point.
1. If point $P$ moves to $(1,3)$, find the equation of tangent line $l$ at this time.
2. Find the trajectory equation of point $P$ that satisfies the condition $|PM|=|PO|$.

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