Functions with operation (multiplication) as argument

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Suppose I have functions $f$, $g_1$, $g_2$, $h$ and an operation $*$, which is not necessarily multiplication, satisfying the following relations:

$$f(X * Y) = g_1(X) g_2(Y) + g_1(Y) g_2(X) + f(X) * f(Y) + h(X * Y)$$

$$h(X * Y) = h^2(X) * g_1(Y)$$

How to find a symbolic expression for $f(X^7)$?