16. The schematic diagram of a computer device is shown in the figure, where $J_{1}, J_{2}$ represent data inputs, and $C$ is the output for the computation result. The computation process involves inputting natural numbers $m$ and $n$ from $J_{1}, J_{2}$ respectively, and after computation, a natural number $k$ is output from $C$. If this device satisfies the following three properties:
(1) If $J_{1}, J_{2}$ both input 1, the output result is 1;
(2) If $J_{1}$ inputs any fixed natural number unchanged, and $J_{2}$ inputs a natural number increased by 1, then the output result is increased by 2;
(3) If $J_{2}$ inputs 1, and $J_{1}$ inputs a natural number increased by 1, then the output result is doubled. Try to answer: (1) If $J_{1}$ inputs 1, and $J_{2}$ inputs a natural number $n$, what is the output result?
(2) If $J_{2}$ inputs 1, and $J_{1}$ inputs a natural number $m$, what is the output result?
(3) If $J_{1}$ inputs the natural number 2002, and $J_{2}$ inputs the natural number 9, what is the output result?
16. When $J_{1}$ inputs $m, J_{2}$ inputs $n$, the output result is denoted as $f(m, n)$. Let $f(m, n)=k$, then $f(1,1)=1, f(m, n+1)=$ $f(m, n)+2, f(m+1,1)=2 f(m, 1)$.
( I ) Since $f(1, n+1)=f(1, n)+2$, it follows that $f(1,1), f(1,2), \cdots, f(1, n) \cdots$ form an arithmetic sequence with $f(1,1)$ as the first term and 2 as the common difference.
Therefore, $f(1, n)=f(1,1)+2(n-1)=2 n-1$.
( II ) Since $f(m+1,1)=2 f(m, 1)$, it follows that $f(1,1), f(2,1), \cdots, f(m, 1), \cdots$ form a geometric sequence with $f(1,1)$ as the first term and 2 as the common ratio.
Therefore, $f(m, 1)=f(1,1) \cdot 2^{m-1}=2^{m-1}$.
(III ) Since $f(m, n+1)=f(m, n)+2$, it follows that $f(m, 1), f(m, 2), \cdots, f(m, n) \cdots$ form an arithmetic sequence with $f(m, 1)$ as the first term and 2 as the common difference.
Therefore, $f(m, n)=f(m, 1)+2(n-1)=2^{m-1}+2 n-2, f(2002,9)=2^{2001}+16$.