
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 system, point $A$ is on the parabola $y=x^{2}$ in the second quadrant. A line $OB$ is drawn perpendicular to $OA$ and intersects the parabola at point $B$. A rectangle $A O B C$ is formed with $OA$ and $OB$ as sides. Given that the x-coordinate of point $A$ is $-\frac{1}{2}$, find the equation of the hyperbola that passes through point $C$.

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