Context Free Grammar: at least 2 1's, at most 2 0's Been having some trouble trying to come up with a CFG for this language: all binary strings that contain least 2 1's and at most 2 0's

So far, I've come up with this:

S --> T | 0T | T0 | T0T | 00TT | TT00 | 0T0T | T0T0 | 0TT0 | T00T

T--> 1T | V

V--> 11Z | 1Z1 | Z11 

Z--> 1Z | epsilon

I realize this is mostly likely incorrect/redundant, so any feedback would be extremely helpful. Thank you!
Hint

If $L_{i j}$ is the language with at least $i$ $1$s and at most $j$ $0$s 

$$\begin{align}
L_{2 2} &amp;= 1 \, L_{1 2} \, | \,?\\
... \\
L_{0 0} &amp;= \,?
\end{align}$$



Edited to answer the correct question.