
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 $A = (-4, 0),$ $B = (-3, 2),$ $C = (3, 2),$ and $D = (4, 0).$ Suppose that point $P$ satisfies \[PA + PD = 10,\] and \[PB + PC = 8.\] Find the $y-$coordinate of $P$, which can be expressed in the form $\frac{-a + b \sqrt{c}}{d},$ where $a, b, c, d$ are positive integers. Find $a + b + c + d.$

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