Consider a polynomial of degree $3m$ such that

$P(0)=P(3)=P(6)...P(3m) = 2$

$P(1)=P(4)=P(7)...P(3m-2) = 1$

$P(2)=P(5)=P(8)...P(3m-1) = 0$

and $P(3m+1) = 6562$

Find the value of $m$?