
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 foci on the $x$-axis and the distance between its two vertices is $2$, the distance from a focus to an asymptote is $\sqrt{2}$.

$(1)$ Find the standard equation of the hyperbola;

$(2)$ Write down the length of the real axis, the length of the imaginary axis, the coordinates of the foci, the eccentricity, and the equations of the asymptotes.

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