
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
Through any point \( P \) on the ellipse \( C: \frac{x^2}{3} + \frac{y^2}{2} = 1 \), a perpendicular line to the directrix of \( C \) is drawn and intersects the ellipse at \( H \) (the foot of the perpendicular). Extend \( PH \) to point \( Q \) such that \( HQ = \lambda PH \) for \( \lambda \geq 1 \). As point \( P \) moves along \( C \), find the range of eccentricities of the locus of point \( Q \).

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