Suppose $z$ and $w$ are complex numbers such that
\[|z| = |w| = z \overline{w} + \overline{z} w= 1.\]Find the largest possible value of the real part of $z + w.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

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