
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 a hyperbola $C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ (a > 0, b > 0) with an eccentricity of $\sqrt{5}$ and a conjugate axis length of 4.
(I) Find the standard equation of the hyperbola;
(II) A line $l$ passing through point (0, 1) with an angle of 45° intersects hyperbola $C$ at points $A$ and $B$. $O$ is the origin. Find the area of triangle $OAB$.

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