Example 24 (2004 Hunan Province Competition Question) The schematic diagram of a computer device is shown in Figure 19, $J_{1} 、 J_{2}$ represent data inputs, $C$ is the output for the calculation result. The calculation process involves inputting natural numbers $m$ and $n$ from $J_{1}$ and $J_{2}$ respectively, and after calculation, a natural number $k$ is output from $C$. If this device satisfies the following three properties:
(1) When 1 is input to both $J_{1}$ and $J_{2}$, the output result is 1;
(2) If any fixed natural number is input to $J_{1}$ and the natural number input to $J_{2}$ increases, then the output result increases by 2;
(3) If 1 is input to $J_{2}$ and the natural number input to $J_{1}$ increases by 1, then the output result is doubled. Try to answer: (1) If 1 is input to $J_{1}$ and a natural number $n$ is input to $J_{2}$, what is the output result?
(2) If 1 is input to $J_{2}$ and a natural number $m$ is input to $J_{1}$, what is the output result?
(3) If 2002 is input to $J_{1}$ and 9 is input to $J_{2}$, what is the output result?
We will change the problem to be expressed in mathematical symbols:
When $J_{1}$ inputs $m$ and $J_{2}$ inputs $n$, the output result is denoted as $f(m, n)$. Let $f(m, n)=k$, then $f(1,1)=1, f(m, n+1)=$ $f(m, n)+2, f(m+1,1)=2 f(m, 1)$.
(1) Since $f(1, n+1)=f(1, n)+2$, thus $f(1,1), f(1,2), \cdots, f(1, n), \cdots$ form an arithmetic sequence with $f(1,1)$ as the first term and 2 as the common difference.
Therefore, $f(1, n)=f(1,1)+2(n-1)=2 n-1$.
(2) Since $f(m+1,1)=2 f(m, 1)$, thus $f(1,1), f(2,1), \cdots, f(m, 1), \cdots$ form a geometric sequence with $f(1,1)$ as the first term and 2 as the common ratio.
Therefore, $f(m, 1)=f(1,1) \cdot 2^{m-1}=2^{m-1}$.
(3) Since $f(m, n+1)=f(m, n)+2$, thus $f(m, 1), f(m, 2), \cdots, f(m, n), \cdots$ form an arithmetic sequence with $f(m, 1)$ as the first term and 2 as the common difference.
Therefore, $f(m, n)=f(m, 1)+2(n-1)=2^{m-1}+2 n-2, f(2002,9)=2^{2001}+16$.