
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 the ellipse \( C: \frac{y^{2}}{a^{2}} + \frac{x^{2}}{b^{2}} = 1 \) with \( a > b > 0 \) and an eccentricity of \( \frac{1}{2} \). The upper and lower foci are \( F_{1} \) and \( F_{2} \), and the right vertex is \( D \). A line through \( F_{1} \) perpendicular to \( D F_{2} \) intersects the ellipse \( C \) at points \( A \) and \( B \), such that \( |B D| - |A F_{1}| = \frac{8 \sqrt{3}}{39} \).

(1) Find the value of \( |A D| + |B D| \).

(2) Lines tangent to the ellipse at points \( A \) and \( B \) intersect at point \( E \). If \( F_{1} E \) intersects the \( x \)-axis at point \( P \) and \( F_{2} E \) intersects the \( x \)-axis at point \( Q \), find the value of \( |P Q| \).

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