
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 $C$: $\dfrac{x^2}{4} - \dfrac{y^2}{b^2} = 1$ ($b > 0$) with an asymptote $y = \dfrac{\sqrt{6}}{2}x$, and foci $F_1$ and $F_2$, find the value of $|\overrightarrow{PF_1} + \overrightarrow{PF_2}|$, given that point $P$ is on the hyperbola $C$ and $|PF_1| : |PF_2| = 3 : 1$.

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