
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
A line l passes through the origin and intersects the left and right branches of the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) at points A and B respectively. Point F ($-\sqrt{3}$, 0) is the left focus of hyperbola C. Given that $|FA| + |FB| = 4$ and $\vec{FA} \cdot \vec{FB} = 0$, find the equation of hyperbola C.

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