An L-expression is defined as any one of the following:

• a natural number;
• the symbol $A$;
• the symbol $Z$;
• the symbol $S$;
• a pair of L-expressions $u,v$, which is written as $u(v)$.

An L-expression can be transformed according to the following rules:

• $A(x)\to x+1$ for any natural number $x$;
• $Z(u)(v)\to v$ for any L-expressions $u,v$;
• $S(u)(v)(w)\to v(u(v)(w))$ for any L-expressions $u,v,w$.

For example, after applying all possible rules, the L-expression $S(Z)(A)(0)$ is transformed to the number $1$: $$S(Z)(A)(0)\to A(Z(A)(0))\to A(0)\to 1.$$ Similarly, the L-expression $S(S)(S(S))(S(Z))(A)(0)$ is transformed to the number $6$ after applying all possible rules.

Define the following L-expressions:

• $C_0=Z$;
• $C_i=S(C_{i-1})$ for $i\ge 1$;
• $D_i=C_i(S)(S)$.

For natural numbers $a,b,c,d,e$, let $F(a,b,c,d,e)$ denote the result of the L-expression $D_a(D_b)(D_c)(C_d)(A)(e)$ after applying all possible rules.

Find the last nine digits of $F(12,345678,9012345,678,90)$.

Note: it can be proved that the L-expression in question can only be transformed a finite number of times, and the final result does not depend on the order of the transformations.