
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
Define an operation $x \otimes y$ for all natural numbers (including zero) $x, $y, satisfying the following three conditions: 
1. $x \otimes 0 = x + 1$;
2. $0 \otimes (x + 1) = 1 \otimes x$;
3. $(x + 1) \otimes (y + 1) = [x \otimes (y + 1)] \otimes y$.

Find the values of $n \otimes 1$ and $2005 \otimes 2$.

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