
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 \( A \) and \( B \) be the intersection points of the hyperbola \( W: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) with the real axis, and let \( P(0,1) \) be a point outside the hyperbola. The lines \( PA \) and \( PB \) intersect the hyperbola again at points \( C \) and \( D \), respectively. The tangent lines to the hyperbola at \( C \) and \( D \) intersect at \( Q \). If the triangle \(\triangle QCD\) is an equilateral triangle with an area of \(\frac{16 \sqrt{3}}{27}\), then find the equation of the hyperbola \( W \).

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