Trace the Dot!

First, cut the range of $t$ to better understand the vector $\overrightarrow{OP}$ .

$i) -2 \leq t < 0$ $\overrightarrow{OP}=(1-t)\overrightarrow{OA}+(2+t)\overrightarrow{OB}, \overrightarrow{OB}=3-\overrightarrow{OA}$ When $t=-2, \overrightarrow{OP}=3\overrightarrow{OA}$ and when $t=0, \overrightarrow{OP}=\overrightarrow{OA}+2\overrightarrow{OB}$ Thus, when $-2 \leq t < 0$ , trace of the dot $P$ is like the blue line below: $ii) 0 \leq t < 1$ $\overrightarrow{OP}=(1-t)\overrightarrow{OA}+(2-t)\overrightarrow{OB}, \overrightarrow{OB}=1+\overrightarrow{OA}$ When $t=1, \overrightarrow{OP}=\overrightarrow{OB}$ Thus, when $0 \leq t < 1$ , trace of the dot $P$ is like the blue line below: $iii) 1 \leq t < 2$ $\overrightarrow{OP}=(t-1)\overrightarrow{OA}+(2-t)\overrightarrow{OB}, \overrightarrow{OB}=1-\overrightarrow{OA}$ When $t=2, \overrightarrow{OP}=\overrightarrow{OA}$ Thus, when $1 \leq t < 1$ , trace of the dot $P$ is like the blue line below: Therefore, the whole trail of dot $P$ is like the blue line below: $A'(-12,0,0), B'(3,2,2), E(2,4,4)$

Finally, find the length of $L$ . ${\overline{A'E}}^2=228=a, {\overline{EB}}^2=9=b^2, {\overline{BA}}^2=57=c$ $a+b^2+c=294$ .