
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 the asymptotes of a hyperbola are $y=\pm \sqrt {2}x$, and it passes through the point $P(2, \sqrt {2})$.

$(1)$ Find the standard equation of the hyperbola;  
$(2)$ A line $l$ that passes through the right focus $F_{2}$ of the hyperbola and has a slope angle of $30^{\circ}$ intersects the hyperbola at points $A$ and $B$. Find $|AB|$.

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