
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 two lines $l\_1$: $2x-y=0$ and $l\_2$: $x+y+2=0$.
(1) Find the equation of the line $l$ that passes through point $P(1,1)$ and is perpendicular to $l\_1$;
(2) If the center of circle $M$ lies on line $l\_1$, is tangent to the $y$-axis, and the chord length intercepted by line $l\_2$ is $\sqrt{2}$, find the equation of circle $M$.

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