
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
In the Cartesian coordinate plane \(xOy\), given that points \(A\) and \(B\) lie on the hyperbola \(C: 2x^{2} + 4x - y^{2} = 0\), and \(\triangle ABC\) is an isosceles right triangle with \(O\) as the right-angle vertex. Find the total number of such \(\triangle OAB\).

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