Build a product-state joint distribution

Using only single-qubit gates from $\{R_X,R_Y,R_Z,H,X\}$ (no two-qubit gates, at most four gates), prepare a two-qubit state whose computational-basis distribution is

$P(00)=\tfrac38,\quad P(01)=\tfrac38,\quad P(10)=\tfrac18,\quad P(11)=\tfrac18 .$

Basis labels are $\lvert q_0q_1\rangle$ with index $k=2q_0+q_1$: qubit 0 is the leftmost bit. Each probability is graded with absolute tolerance $10^{-6}$; the grader also checks the equivalent single-qubit expectations $\langle Z\otimes I\rangle=P_{q_0}(0)-P_{q_0}(1)$ and $\langle I\otimes Z\rangle=P_{q_1}(0)-P_{q_1}(1)$.