
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 $C\_1$: $x^{2}+y^{2}=2$ and circle $C\_2$, line $l$ is tangent to circle $C\_1$ at point $(1,1)$; the center of circle $C\_2$ is on the ray $2x-y=0 (x\geqslant 0)$, circle $C\_2$ passes through the origin, and the length of the chord intercepted by line $l$ on circle $C\_2$ is $4\sqrt{3}$.
(1) Find the equation of line $l$;
(2) Find the equation of circle $C\_2$.

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