
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
Consider a hyperbola $C$ with the origin $O$ as its center, and its foci located on the x-axis. One of the asymptotes of $C$ has an inclination angle of $60^\circ$. Point $F$ is the right focus of the hyperbola $C$. Point $M$ lies on the hyperbola $C$ in the first quadrant, and point $N$ is the midpoint of the line segment $MF$. If $|\overrightarrow{ON}| = |\overrightarrow{NF}| + 1$, derive the equation of hyperbola $C$.

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