
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]


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Given that the center of a hyperbola is at the origin, its foci \\(F_1\\) and \\(F_2\\) are on the coordinate axes, and one of its asymptotes is \\(y=x\\), and it passes through the point \\((4, -\sqrt{10})\\).  
\\((1)\\) Find the equation of the hyperbola;  
\\((2)\\) If point \\(M(3,m)\\) is on this hyperbola, find \\( \overrightarrow{MF_{1}} \cdot \overrightarrow{MF_{2}}\\).

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