
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 hyperbola \(C: x^{2}+2x-my^{2}=0\) (with \(m>0\)) that passes through the origin \(O\), determine if there exist two points \(A\) and \(B\) on \(C\) such that \(\triangle AOB\) is an isosceles right triangle with \(O\) as the right angle vertex. If no such points exist, explain why; if such points do exist, determine how many distinct isosceles right triangles there are.

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