
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: $$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$$ (a > 0, b > 0) with the same asymptotes as the hyperbola $$\frac{y^{2}}{6} - \frac{x^{2}}{2} = 1$$ and passes through point (2, 3).
(1) Find the equation of hyperbola C;
(2) Given that the left and right foci of hyperbola C are $F_1$ and $F_2$, respectively, a line $l$ passing through $F_2$ with an angle of inclination of $\frac{3}{4}π$ intersects hyperbola C at points A and B. Find the area of triangle $F_1AB$.

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