
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
Given $N$ blue sticks and $N$ red sticks, the sum of the lengths of the blue sticks is equal to the sum of the lengths of the red sticks. It is known that the blue sticks can form an $N$-gon and the red sticks can also form an $N$-gon. Can you always choose one blue stick and one red stick and swap their colors (turn the blue one red and the red one blue) so that the blue sticks can still form an $N$-gon and the red sticks can also still form an $N$-gon?

Solve the problem

a) for $N = 3$

b) for any natural number $N > 3$.

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