Point $B$ is due east of point $A$. Point $C$ is due north of point $B$. The distance between points $A$ and $C$ is $10\sqrt{2}$, and $\angle BAC = 45^\circ$. Point $D$ is $20$ meters due north of point $C$. Calculate the distance $AD$ and determine the interval in which this distance falls, expressed as (a, b) where a and b are consecutive integers. Please find the value of a + b.

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