
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 triangle $\triangle ABC$, the coordinates of points $A$ and $B$ are $A(-\sqrt{2}, 0)$ and $B(\sqrt{2}, 0)$, respectively, and point $C$ is located above the $x$-axis.
(1) If the coordinate of point $C$ is $(\sqrt{2}, 1)$, find the equation of the ellipse with foci at $A$ and $B$ and passing through point $C$.
(2) A line $l$ with a slope of $\frac{3\pi}{4}$ passes through point $P(m, 0)$ and intersects the curve from part (1) at points $M$ and $N$. If point $Q(1, 0)$ lies on the circle with $\overline{MN}$ as the diameter, find the value of the real number $m$.

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