CF505C Mr. Kitayuta, the Treasure Hunter

The Shuseki Islands are an archipelago of $ 30001 $ small islands in the Yutampo Sea. The islands are evenly spaced along a line, numbered from $ 0 $ to $ 30000 $ from the west to the east. These islands are known to contain many treasures. There are $ n $ gems in the Shuseki Islands in total, and the $ i $ -th gem is located on island $ p_{i} $ . Mr. Kitayuta has just arrived at island $ 0 $ . With his great jumping ability, he will repeatedly perform jumps between islands to the east according to the following process: - First, he will jump from island $ 0 $ to island $ d $ . - After that, he will continue jumping according to the following rule. Let $ l $ be the length of the previous jump, that is, if his previous jump was from island $ prev $ to island $ cur $ , let $ l=cur-prev $ . He will perform a jump of length $ l-1 $ , $ l $ or $ l+1 $ to the east. That is, he will jump to island $ (cur+l-1) $ , $ (cur+l) $ or $ (cur+l+1) $ (if they exist). The length of a jump must be positive, that is, he cannot perform a jump of length $ 0 $ when $ l=1 $ . If there is no valid destination, he will stop jumping. Mr. Kitayuta will collect the gems on the islands visited during the process. Find the maximum number of gems that he can collect.

N/A

N/A

In the first sample, the optimal route is 0 $ → $ 10 (+1 gem) $ → $ 19 $ → $ 27 (+2 gems) $ →...
$ In the second sample, the optimal route is 0 $ → $ 8 $ → $ 15 $ → $ 21 $ → $ 28 (+1 gem) $ → $ 36 (+1 gem) $ → $ 45 (+1 gem) $ → $ 55 (+1 gem) $ → $ 66 (+1 gem) $ → $ 78 (+1 gem) $ →... $ In the third sample, the optimal route is 0 $ → $ 7 $ → $ 13 $ → $ 18 (+1 gem) $ → $ 24 (+2 gems) $ → $ 30 (+1 gem) $ →... $