
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 $A = (-3, 0),$ $B=(-3,2),$ $C=(3,2),$ and $D=(3,0).$ Suppose that point $P$ satisfies 
\[PA + PD = PB + PC = 10.\]
Determine the $y-$coordinate of $P$ such that it 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}
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