Solve in $\Bbb{N}^*$ the equation $$ 4^a \cdot 5^b - 3^c \cdot 11^d = 1.$$
**
   \[
   4^1 \cdot 5^2 - 3^2 \cdot 11^1 = 4 \cdot 25 - 9 \cdot 11 = 100 - 99 = 1
   \]
   The solution \( (a, b, c, d) = (1, 2, 2, 1) \) satisfies the equation.

The final answer is \( \boxed{ (a, b, c, d) = (1, 2, 2, 1) } \)