
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
Juquinha places points on a circumference and draws segments connecting some of these points. He calls a point an odd-point if it is connected to an odd number of points, and an even-point otherwise. For example, in the illustration, he chose five points and made four connections.

a) Juquinha marked five points on a circumference and drew all possible connections except one. How many odd-points were obtained?

b) Juquinha marked six points on each of the circumferences below. In each case, show how to obtain the indicated number of odd-points with exactly five connections.

c) Explain why Juquinha will always find an even number of odd-points, regardless of the number of points he marks or the number of connections he makes.

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