
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 $C: \frac{x^{2}}{4} - \frac{y^{2}}{b^{2}}=1 (b > 0)$ with one of its asymptotes defined by the equation $y= \frac{\sqrt{6}}{2}x$, and $F_{1}, F_{2}$ representing the left and right foci of hyperbola $C$, respectively. Let $P$ be a point on hyperbola $C$ such that the ratio of distances from $P$ to the foci is $|PF_{1}|:|PF_{2}|=3:1$. Determine the value of $|\overrightarrow{PF_{1}}+ \overrightarrow{PF_{2}}|$.

A) $4$
B) $2\sqrt{6}$
C) $2\sqrt{10}$
D) $\frac{6\sqrt{10}}{5}$

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