
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 $C$ with foci $F_{1}$ and $F_{2}$, and eccentricity $\sqrt{5}$, let $P$ be a point on $C$ such that $2|PF_{1}|=3|PF_{2}|$. If the area of $\triangle PF_{1}F_{2}$ is $2\sqrt{5}$, calculate the length of the real axis of hyperbola $C$.

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