
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
Kevin is in kindergarten, so his teacher puts a  $100 \times 200$  addition table on the board during class. The teacher first randomly generates distinct positive integers  $a_1, a_2, \dots, a_{100}$  in the range  $[1, 2016]$  corresponding to the rows, and then she randomly generates distinct positive integers  $b_1, b_2, \dots, b_{200}$  in the range  $[1, 2016]$  corresponding to the columns. She then fills in the addition table by writing the number  $a_i+b_j$  in the square  $(i, j)$  for each  $1\le i\le 100$ ,  $1\le j\le 200$ .

During recess, Kevin takes the addition table and draws it on the playground using chalk. Now he can play hopscotch on it! He wants to hop from  $(1, 1)$  to  $(100, 200)$ . At each step, he can jump in one of  $8$  directions to a new square bordering the square he stands on a side or at a corner. Let  $M$  be the minimum possible sum of the numbers on the squares he jumps on during his path to  $(100, 200)$  (including both the starting and ending squares). The expected value of  $M$  can be expressed in the form  $\frac{p}{q}$  for relatively prime positive integers  $p, q$ . Find  $p + q.$ *Proposed by Yang Liu*

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