
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
For any point $P$ on the ellipse $C: \frac{x^2}{3} + \frac{y^2}{2} = 1$, draw a perpendicular line $PH$ from $P$ to the right directrix of the ellipse $C$ (where $H$ is the foot of the perpendicular). Extend $PH$ to point $Q$ such that $|HQ| = \lambda |PH|$ ($\lambda \geq 1$). When point $P$ moves on the ellipse $C$, determine the range of the eccentricity of the trajectory of point $Q$.

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