Closed form of infinite product $ \prod\limits_{k=0}^\infty \left(1+\frac{1}{2^{2^k}}\right)$

What will be the value of the following Infinite Product :

$$\displaystyle \prod_{k=0}^\infty \left(1+\dfrac{1}{2^{2^k}}\right)$$

It would be nice if anyone could spare the time and boil down to the absolute basics and tell how they reached the solution.

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