
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 $C: x^{2}+y^{2}+2x-4y+m=0$.  
$(1)$ If the chord of circle $C$ on the $x$-axis is equal to half the radius, find the value of $m$;  
$(2)$ When $m=3$, if the tangent of circle $C$ has equal intercepts on the $x$-axis and $y$-axis, find the equation of this tangent.

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