
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. \( J_1 \) and \( J_2 \) represent data inputs, and \( C \) is the output for the calculation result. The calculation process involves inputting natural numbers \( m \) and \( n \) into \( J_1 \) and \( J_2 \) respectively, after which a natural number \( k \) is output by \( C \). This device satisfies the following three properties:
1. If both \( J_1 \) and \( J_2 \) input 1, the output is 1.
2. If the input to \( J_1 \) is a fixed natural number and the input to \( J_2 \) increases, the output increases by 2.
3. If \( J_2 \) inputs 1 and the input to \( J_1 \) increases by 1, the output is doubled.
Answer the following:
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}
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