
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
Five points $O, A, B, C, D$ are placed sequentially on a straight line so that $OA = 2a$, $OB = b$, $OC = c$, and $OD = 2d$, where $a, b, c, d$ are positive numbers. $P$ is a point on the line between $B$ and $C$. We are given that $AP:PD = 2(BP:PC)$. Find the expression for $OP$.
**A)** $\frac{2a - c + 2b - 2d}{2}$   
**B)** $\frac{4a - 2c + 4b - 4d}{2}$   
**C)** $\frac{(2a - c + 2b - 2d) + \sqrt{(2a - c + 2b - 2d)^2 - 8(ac - bd)}}{2}$   
**D)** $\frac{(2a - c + 2b - 2d) - \sqrt{(2a - c + 2b - 2d)^2 - 8(ac - bd)}}{2}$

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