
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A schematic diagram of a computer device is shown, where $J_1$ and $J_2$ represent data inputs, and $C$ is the output for calculation results. The calculation process involves inputting natural numbers $m$ and $n$ into $J_1$ and $J_2$, respectively, and outputting a natural number $k$ from $C$ after computation. This device satisfies the following three properties:
1. If $J_1$ and $J_2$ both input 1, the output is 1.
2. If $J_1$ inputs any fixed natural number and $J_2$ inputs a natural number increased by 1, the output increases by 2.
3. If $J_2$ inputs 1, and $J_1$ inputs a natural number increased by 1, the output becomes twice the original.

Questions:
1. If $J_1$ inputs 1 and $J_2$ inputs a natural number $n$, what is the output?
2. If $J_2$ inputs 1 and $J_1$ inputs a natural number $m$, what is the output?
3. If $J_1$ inputs the natural number 2002 and $J_2$ inputs the natural number 9, what is the output?

Present the answer in LaTex format: \boxed{Your answer}
user