
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 an ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, its minor axis is half the length of its major axis, and the maximum distance from a point on the ellipse $C$ to its right focus $F$ is $2 + \sqrt{3}$. The line $L$: $y = x + m$ intersects the ellipse $C$ at two distinct points $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$.

(1) Find the equation of the ellipse $C$;

(2) If the area of $\triangle AOB$ is $1$, find the equation of the line $L$.

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