If vector $\overrightarrow a$ is unit vector and it makes angle $\large\frac{\pi}{4}$ with $\hat k$ and $\overrightarrow a+\hat i+\hat j$ is also a unit vector, then $\overrightarrow a=?$

Question

$\begin{array}{1 1} (A) \large\frac{1}{2}\hat i+\large\frac{1}{2}+\hat j+\large\frac{1}{\sqrt 2}\hat k \\ (B) -\large\frac{1}{2}\hat i+\large\frac{1}{2}+\hat j+\large\frac{1}{\sqrt 2}\hat k\\ (C) -\large\frac{1}{2}\hat i-\large\frac{1}{2}+\hat j+\large\frac{1}{\sqrt 2}\hat k\\ (D) \large\frac{1}{2}\hat i-\large\frac{1}{2}+\hat j+\large\frac{1}{\sqrt 2}\hat k \end{array}$