
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 \( A = (-4, 0), B = (-1, 1), C = (1, 1), D = (4, 0) \). Suppose that point \( P \) satisfies the conditions \[ PA + PD = PB + PC = 10. \] Determine the \( y \)-coordinate of \( P \), which should be represented as \(\frac{-a + b \sqrt{c}}{d}\), where \( a, b, c, d \) are positive integers. Find \( a + b + c + d \).

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