
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 $E$ passes through points $A(0,0)$ and $B(1,1)$, choose one of the following $3$ conditions:<br/>① Passes through point $C(2,0)$;<br/>② Circle $E$ is always bisected by the line $mx-y-m=0$ where $m\in R$;<br/>③ Tangent to the $y$-axis.<br/>$(1)$ Find the equation of circle $E$;<br/>$(2)$ The line $l$ passing through point $P(2,3)$ is tangent to circle $E$, find the equation of line $l$.

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