Please reason step by step, and put your final answer within \boxed{}.
Given two unit vectors $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ with an angle of $\frac{\pi}{3}$ between them, if vector $\overrightarrow{b_{1}}= \overrightarrow{e_{1}}-2 \overrightarrow{e_{2}}$ and $\overrightarrow{b_{2}}= \overrightarrow{e_{1}}+4 \overrightarrow{e_{2}}$, then find the magnitude of $\overrightarrow{b_{1}}+ \overrightarrow{b_{2}}$.
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