
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 that circle $C$ passes through points $A(2,0)$ and $B(0,4)$, and the center of the circle lies on the line $x+2y-9=0$.
$(1)$ Find the standard equation of circle $C$.
$(2)$ A line $l$ passing through point $P(-2,8)$ is tangent to circle $C$. Find the equation of line $l$.

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