
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 asymptote lines of the hyperbola $C$ are $y=±x$, and one of its semi-axes is $x= \frac{ \sqrt{2}}{2}$.
(1) Find the equation of the hyperbola $C$;
(2) Let line $l$ pass through point $M(-2,0)$ and intersect hyperbola $C$ at points $A$ and $B$. If the area of triangle $OAB$ is $2 \sqrt{3}$, find the equation of line $l$;
(3) In part (2), does there exist a line $l$ such that $OA \perp OB$? If it exists, find the equation of line $l$; if not, explain why.

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