HMMT 2007 U35

The Algorithm. There are thirteen broken computers situated at the following set $S$ of thirteen points in the plane:

$$\begin{array}{lll} A=(1,10) & B=(976,9) & C=(666,87) \\ D=(377,422) & E=(535,488) & F=(775,488) \\ G=(941,500) & H=(225,583) & I=(388,696) \\ J=(3,713) & K=(504,872) & L=(560,934) \\ & M=(22,997) & \end{array}$$

At time $t=0$, a repairman begins moving from one computer to the next, traveling continuously in straight lines at unit speed. Assuming the repairman begins and $A$ and fixes computers instantly, what path does he take to minimize the total downtime of the computers? List the points he visits in order. Your score will be $\left\lfloor\frac{N}{40}\right\rfloor$, where

$$N=1000+\lfloor\text { the optimal downtime }\rfloor-\lfloor\text { your downtime }\rfloor,$$

or 0 , whichever is greater. By total downtime we mean the sum

$$\sum_{P \in S} t_{P}$$

where $t_{P}$ is the time at which the repairman reaches $P$.