  We construct a gauge-mediated SUSY breaking model with vector-like matters
combined with the Peccei-Quinn mechanism to solve the strong CP problem. The
Peccei-Quinn symmetry plays an essential role for generating sizable masses for
the vector-like matters and the $\mu$-term without introducing dangerous CP
angle. The model naturally explains both the 125GeV Higgs mass and the muon
anomalous magnetic moment. The stabilization of the Peccei-Quinn scalar and the
cosmology of the saxion and axino are also discussed.
