
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
It is known that the center of circle $C_1$ is at the origin $O$, and it is tangent to the line $l_1: x - y - 2\sqrt{2} = 0$.
(1) Find the standard equation of the circle;
(2) Let point $A(x_0, y_0)$ be any point on the circle, and $AN \perp x$-axis at $N$. If the moving point $Q$ satisfies $\overrightarrow{OQ} = m\overrightarrow{OA} + n\overrightarrow{ON}$, (where $m + n = 1$, $m, n \neq 0$, $m$ is a constant), try to find the trajectory equation $C_2$ of the moving point $Q$.

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