
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 with center $C$ and equation $x^{2}+y^{2}+2x-4y+m=0$, which intersects with the line $2x+y-3=0$ at points $A$ and $B$:

$(1)$ If $\triangle ABC$ is an equilateral triangle, find the value of $m$;  
$(2)$ Is there a constant $m$ such that the circle with diameter $AB$ passes through the origin? If so, find the value of $m$; if not, explain why.

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