
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 $C_{1}:{x}^{2}+{y}^{2}-2x-4y+4=0$ intersects with the line $l:x+2y-4=0$ at points $A$ and $B$. <br/>$(1)$ Find the length of chord $AB$; <br/>$(2)$ If circle $C_{2}$ passes through points $E\left(1,-3\right)$ and $F\left(0,4\right)$, and the line containing the common chord of circle $C_{2}$ and circle $C_{1}$ is parallel to the line $2x+y+1=0$, find the equation of circle $C_{2}$.

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