It gets a bit weird

a12+a22++a192=a202++a372a_1^2+a_2^2+\cdots +a_{19}^2=a_{20}^2+\cdots +a_{37}^2

Do there exist consecutive integers a1,,a37a_1,\ldots ,a_{37} such the equation above holds? If so, enter the largest possible value of a1a_1. If you come to the conclusion that no such integers exist, enter 999.

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