
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 hyperbola with its center at the origin, and its foci $F_1$ and $F_2$ are on the coordinate axes, the eccentricity is $\sqrt{2}$, and it passes through the point $(4, -\sqrt{10})$. Point $M(3, m)$ is on the hyperbola.
(1) Find the equation of the hyperbola;
(2) Find the area of $\triangle F_1MF_2$.

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