
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
Let the hyperbola $C$: $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt 0,b \gt 0\right)$ have left and right foci $F_{1}$ and $F_{2}$ respectively, with eccentricity $\sqrt{5}$. $P$ is a point on $C$ such that $F_{1}P\bot F_{2}P$. If the area of $\triangle PF_{1}F_{2}$ is $4$, find the value of $a$.

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