
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

Professor Carlão decided to create a math problem worth a total of 10 points, which consists of three parts: \(a\), \(b\), and \(c\). After creating the parts, he was unsure how best to distribute the 10 points among the parts such that each part is worth a positive integer number of points.

a) Joana, a colleague of Carlão, suggested that part \(c\) should be worth the same number of points as the sum of parts \(a\) and \(b\) because, according to her, part \(c\) is more difficult. If Carlão follows Joana's suggestion, how many different ways can he distribute the points?

b) Disregarding Joana's suggestion, that is, considering that Carlão will distribute the points in any manner, how many different ways can he distribute the 10 points among the three parts?

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