
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 foci of a hyperbola are on the x-axis, the distance between the two vertices A<sub>1</sub> and A<sub>2</sub> is 2, and the distance from a focus to an asymptote is $\sqrt{2}$,
(1) Find the standard equation of the hyperbola;
(2) Let the coordinates of an arbitrary point (different from the two vertices) on the hyperbola be M(x, y). The slopes of the lines MA<sub>1</sub> and MA<sub>2</sub> are $k_1$ and $k_2$, respectively. Find the value of $k_1k_2$.

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