
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 $C$ passing through points $A(-1,1)$ and $B(-2,-2)$, and the center of the circle lies on the line $l: x+y-1=0$, find:<br/>
$(1)$ Find the standard equation of the circle with center $C$;<br/>
$(2)$ Let point $P$ be on circle $C$ and point $Q$ be on the line $x-y+5=0$, find the minimum value of $|PQ|$;<br/>
$(3)$ If a tangent line to the circle $C$ passes through the point $(0,3)$, find the equation of this tangent line.

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