
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 $C$ with its center on the line $x+y+1=0$ and a radius of $5$, and the circle $C$ passes through points $P(-2,0)$ and $Q(5,1)$.  
$(1)$ Find the standard equation of circle $C$;  

$(2)$ Find the equation of the tangent line passing through point $A(-3,0)$ and tangent to circle $C$.

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