
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
Kiara wrote 20 integer numbers on a blackboard, all of them different from zero. Next, for each pair of numbers written by Kiara, Yndira wrote on the same board the respective product between them (including if the result of a product was already written, Yndira repeated it).

For example, if the numbers 2, 3, 4, and 6 were among those written by Kiara, then Yndira would have written the numbers 6, 8, 12, 12, 18, and 24, because \(6 = 2 \times 3\), \(8 = 2 \times 4\), \(12 = 2 \times 6 = 3 \times 4\), \(18 = 3 \times 6\), and \(24 = 4 \times 6\). Note that the number 6 would have been written again even though it was already written by Kiara, while the number 12 would have been written twice by Yndira.

a) In total, how many numbers were written on the blackboard by Kiara and Yndira?

b) Suppose that, out of the total numbers written on the board, exactly 120 are positive. If Kiara wrote more positive numbers than negative, determine how many of the numbers written by Kiara were positive.

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