The Fibonacci sequence modulo 3 is defined by $F_n \equiv F_{n-1} + F_{n-2} \pmod{3}$ with $F_0 \equiv 0, F_1 \equiv 1 \pmod{3}$. Compute the sequence and determine the period (the smallest positive $p$ such that $(F_p, F_{p+1}) \equiv (F_0, F_1) \pmod{3}$).