Both $[Co(NH_3)_6]^{3+}$ and $[CoF_6]^{3-}$ have the central metal ion $Co$ in the +3 oxidation state, which has a $3d^6$ electronic configuration. In $[Co(NH_3)_6]^{3+}$, $NH_3$ acts as a strong field ligand, causing the pairing of $3d$ electrons against Hund's rule. This results in a $t_{2g}^6 e_g^0$ configuration with no unpaired electrons. It undergoes $d^2sp^3$ hybridization, forming an inner-orbital octahedral complex that is diamagnetic. In $[CoF_6]^{3-}$, $F^-$ is a weak field ligand and does not cause the pairing of $3d$ electrons. It has a $t_{2g}^4 e_g^2$ configuration with four unpaired electrons. It undergoes $sp^3d^2$ hybridization, forming an outer-orbital octahedral complex that is highly paramagnetic. Therefore, both complexes are octahedral but have different magnetic behaviours, making both Statement I and Statement II true.