
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 the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \; (a, b>0)$ with left and right foci at $F_{1}(-c, 0)$ and $F_{2}(c, 0)$, respectively, and an eccentricity of 2. If the line $y=2(x-c)$ intersects the hyperbola at points $A$ and $B$, and the incenters of $\triangle A F_{1} F_{2}$ and $\triangle B F_{1} F_{2}$ are $I_{1}$ and $I_{2}$ respectively, and $\left|I_{1} I_{2}\right|=2 \sqrt{5}$, find the value of $a$.

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