
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
Five blue candies and four yellow candies can be combined to make many different flavors. Flavors are considered the same if the ratio of blue to yellow candies is the same, so 5 blue / 0 yellow is the same flavor as 1 blue / 0 yellow; and 5 blue / 2 yellow is the same flavor as 1 blue / 0.4 yellow. If a flavor is to be made using some or all of the nine candies, how many different flavors are possible?

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