
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]


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Given that the foci of the ellipse C are $${{F}_{1}}(-1,0),{{F}_{2}}(1,0)$$, and a line passing through F₂ intersects C at points A and B. If $$|A{{F}_{2}}|=2|{{F}_{2}}B|$$ and $$|AB|=|B{{F}_{1}}|$$, find the equation of C.

A) $$\frac{{{x}^{2}}}{2}+{{y}^{2}}=1$$
B) $$\frac{{{x}^{2}}}{3}+\frac{{{y}^{2}}}{2}=1$$
C) $$\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{3}=1$$
D) $$\frac{{{x}^{2}}}{5}+\frac{{{y}^{2}}}}{4}=1$$

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