
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let  $S$  be the set of all lattice points  $(x, y)$  in the plane satisfying  $|x|+|y|\le 10$ . Let  $P_1,P_2,\ldots,P_{2013}$  be a sequence of 2013 (not necessarily distinct) points such that for every point  $Q$  in  $S$ , there exists at least one index  $i$  such that  $1\le i\le 2013$  and  $P_i = Q$ . Suppose that the minimum possible value of  $|P_1P_2|+|P_2P_3|+\cdots+|P_{2012}P_{2013}|$  can be expressed in the form  $a+b\sqrt{c}$ , where  $a,b,c$  are positive integers and  $c$  is not divisible by the square of any prime. Find  $a+b+c$ . (A *lattice point* is a point with all integer coordinates.)
<details><summary>Clarifications</summary>

- $k = 2013$ , i.e. the problem should read, ``... there exists at least one index  $i$  such that  $1\le i\le 2013$  ...''.  An earlier version of the test read  $1 \le i \le k$ .
</details>

*Anderson Wang*

Present the answer in LaTex format: \boxed{Your answer}
user