```
def multiply_optimized(poly1, poly2):
n = max(len(poly1),len(poly2))
if n == 1: #recursion base condition
return multiply_basic(poly1,poly2)
(A0, A1), (B0, B1) = split(poly1, poly2)
Y = polymulti1(add(A0,A1),add(B0,B1))
U = polymulti1(A0,B0)
Z = polymulti1(A1,B1)
second_c = increase_exponent(sub(sub(Y,U),Z),n//2)
second_and_third_sum = add(second_c,increase_exponent(Z,2*(n//2)))
product = add(U,second_and_third_sum)
return product
```

i’d like to know what kind of outputs are expected. For example, if an empty list is given what should my output be? and are there anything else that i should know?