
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 a circle $C$ that is tangent to the line $x+y-2\sqrt{2}=0$, the center of the circle lies on the $x$-axis, and the chord length intercepted by the line $y=x$ on circle $C$ is $4\sqrt{2}$.
(1) Find the equation of circle $C$;
(2) A line $l$ passing through point $M(-1,0)$ with a slope of $k$ intersects circle $C$ at points $A$ and $B$. If the product of the slopes of lines $OA$ and $OB$ is $m$, and $\frac{m}{k^2} = -3-\sqrt{2}$, find the value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$.

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