Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^\circ$, $\angle B = 105^\circ$, $\angle C = 90^\circ$ and $AB = 2$, $BC = CD = DE = \sqrt{2}$. If the length of $AE$ is $\sqrt{a} - b$ where $a$ and $b$ are integers, what is the value of $a + b$?

Let's think step by step and output the final answer within \boxed{}.
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