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

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