
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 a circle $O$ with radius $R$ and an external line $l$. The projection of $O$ onto $l$ is $H$, and $O H = d$. The line segment $O H$ intersects circle $O$ at points $C$ and $D$, with $C H > D H$. Let points $P$ and $Q$ be on $l$, such that $P$ and $Q$ are on the same side of $H$ with $P H = a$ and $Q H = b$, where $a > b$. There are 2008 chords $A_{i} B_{i}(i=1,2, \cdots, 2008)$ in circle $O$ which are parallel to $l$ and their intersection points with $C D$ divide $C D$ evenly. Then
$$
\frac{1}{2008} \sum_{i=1}^{2008}\left(P A_{i}^{2}+P B_{i}^{2}+Q A_{i}^{2}+Q B_{i}^{2}\right)
$$
is expressed in terms of $a, b, d,$ and $R$.

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