
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 circle $C_{1}$: $\left(x-1\right)^{2}+\left(y-1\right)^{2}=1$, circle $C_{2}$: $x^{2}+y^{2}-8x-2y+7=0$.<br/>$(1)$ If the line $l$ passes through the common points of circles $C_{1}$ and $C_{2}$, find the equation of line $l$;<br/>$(2)$ If circle $C_{3}$ passes through the intersection points of the two circles and its center $C_{3}$ lies on the line $y=x$, find the equation of circle $C_{3}$.

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