
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 that the axes of symmetry of ellipse $M$ are the coordinate axes, and one focus of the parabola $y^{2}=4x$ is also a focus of ellipse $M$. A circle with the radius equal to the length of the minor axis of ellipse $M$ and centered at focus $F$ is tangent to line $l: x - 2\sqrt{2}y + 2 = 0$.

(1) Find the equation of ellipse $M$.
(2) Given that line $y=x+m$ intersects ellipse $M$ at points $A$ and $B$, and there exists a point $P$ on ellipse $M$ such that $\overrightarrow{OP}=\overrightarrow{OA}+\overrightarrow{OB}$, find the value of $m$.

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