
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 an ellipse \(C: \frac{x^{2}}{3}+\frac{y^{2}}{2}=1\) and an arbitrary point \(P\) on it, draw a line perpendicular to the ellipse's right directrix through \(P\), denoted as \(PH\) (where \(H\) is the foot of the perpendicular). Extend \(PH\) to a point \(Q\) such that \(|HQ| = \lambda |PH|\) \(\left(\frac{2}{3} \leqslant \lambda \leqslant 1\right)\). Find the equation of the locus of point \(Q\) as \(P\) moves on the ellipse \(C\), given that the eccentricity range for the trajectory of \(Q\) is \(\left[\frac{\sqrt{3}}{3}, 1\right)\).

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