
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 that the coordinates of the foci of ellipse $C$ are $F\_1(-2,0)$ and $F\_2(2,0)$, and one of the vertices on the minor axis is $B(0,- \sqrt {5})$:
1. Find the standard equation of ellipse $C$;
2. A line passing through $F\_1$ intersects the ellipse at points $A$ and $B$, with a slope of $45$ degrees, find the area of $\triangle ABF\_2$.

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