
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
A machine dispenses candies in two colors: five purple candies and four orange candies. A flavor is considered unique based on the percentage of purple candies compared to orange candies, such that 4 purple / 0 orange is the same flavor as 2 purple / 0 orange; similarly, 5 purple / 3 orange is the same as 5 purple / 4 orange. How many different flavors can be made if some or all of the nine candies are used?

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