
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 that line $l$ intersects the parabola $y^{2}=3x$ at points $A$ and $B$, and $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$ (where $O$ is the origin), let line $l$ intersect the $x$-axis at point $F$. $F'$ and $F$ are the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$). If there exists a point $P$ on the right branch of the hyperbola such that $| \overrightarrow{PF'} | = 2 | \overrightarrow{PF} |$, determine the range of values for $a$.

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