
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 $A(0,1)$ and $B(1,4)$, and ____. Choose one from the following $3$ conditions to fill in the blank, and solve the problem. ① passes through the intersection point $C$ of the lines $x-5y-5=0$ and $x-2y-8=0$ ② circle $E$ is always bisected by the line $l: (m+1)x+(m-3)y-6m-2=0 (m\in R)$ ③ is tangent to the $y$-axis. <br/> $(1)$ Find the equation of circle $E$; <br/> $(2)$ Find the equation of the tangent line to circle $E$ passing through point $P(10,11)$.

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