Let $P(x)=a_1x^{699}+a_2x^{698}+...+a_{699}x+a_{700}$ $$b_k = a_1+a_2+ ... + a_k \implies b_{700} \leq 2022$$$$b_1 < b_2 < ... <b_{700}$$consider the following numbers. $${b_1,b_2,...b_{700}}$$$${b_1+22,...,b_{700}+22}$$$${b_1+77,b_2+77,...,b_{700}+77}$$We are looking at $2100$ numbers, and each of these does not exceed $2099$ because $ b_ {700} +77 \leq 2022 + 77 = 2099 $ hence $i$ and $j$ are present in which the following cases may occur.$i>j$ $1)b_i= b_j+22,2)b_i=b_j+77,3)b_i +22 = b_j+77$ in each case the sum of any sequence of numbers will be 22.55 or 77.$\blacksquare$