
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 straight lines $l_1: 2x - y = 0$ and $l_2: x + y + 2 = 0$.
(1) Find the equation of the straight line $l$ that passes through point $P(1, 1)$ and is perpendicular to $l_1$.
(2) If the center of the circle $M$ lies on $l_1$, the circle 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}
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