Problem #WSP-5125

Problem

If a magician puts $1$ dove into his hat, he pulls out $2$ rabbits and $2$ flowers from it. If the magician puts $1$ rabbit in, he pulls out $2$ flowers and $2$ doves. If he puts $1$ flower in, he pulls out $1$ rabbit and $3$ doves. The magician starts with $1$ rabbit. Could he end up with the same number of rabbits, doves, and flowers after performing his hat trick several times?