
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
A circle \( O \) with radius \( R \) is given. There is a line \( l \) outside the circle. Let the projection of \( O \) on \( l \) be \( H \). The distance \( OH = d \). \( OH \) intersects the circle \( O \) at points \( C \) and \( D \) with \( CH > DH \). Let \( P \) and \( Q \) be points on \( l \), on the same side of \( H \), such that \( PH = a \) and \( QH = b \) with \( a > b \). There are 2008 chords \( A_i B_i \) \((i=1,2,\ldots, 2008)\) in the circle parallel to \( l \), and these 2008 chords divide \( CD \) into equal segments. Find the value of \(\frac{1}{2008} \sum_{i=1}^{2008} (PA_i^2 + PB_i^2 + QA_i^2 + QB_i^2)\) in terms of \( a, b, d, \) and \( R \).

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