
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 $C$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ have its left and right foci as $F_1$ and $F_2$, respectively, with $|F_1F_2| = 2c$. A perpendicular line to the x-axis passing through $F_2$ intersects the hyperbola in the first quadrant at point $A$. Given $Q(c, \dfrac{3a}{2})$ and $|F_2Q| > |F_2A|$, point $P$ is a moving point on the right branch of the hyperbola $C$, and $|PF_1| + |PQ| > \dfrac{3}{2}|F_1F_2|$ always holds. Determine the range of the eccentricity of the hyperbola.

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