It gets a bit weird

Number Theory Level 4

\[a_1^2+a_2^2+\cdots +a_{19}^2=a_{20}^2+\cdots +a_{37}^2\]

Do there exist consecutive integers \(a_1,\ldots ,a_{37}\) such the equation above holds? If so, enter the largest possible value of \(a_1\). If you come to the conclusion that no such integers exist, enter 999.