
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 circle $C$: $x^{2}+y^{2}+2x-4y+3=0$.

$(1)$ If the tangent of circle $C$ has equal intercepts on the $x$-axis and $y$-axis, find the equation of the tangent line;  
$(2)$ From a point $P(x_{1},y_{1})$ outside circle $C$, draw a tangent line to the circle at point $M$ such that $|PM|=|PO|$ (where $O$ is the origin), find the coordinates of point $P$ when $|PM|$ is minimized.

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