
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 group of adventurers is showing their loot. It is known that exactly 4 adventurers have rubies; exactly 10 have emeralds; exactly 6 have sapphires; and exactly 14 have diamonds. Additionally, it is known that:

- If an adventurer has rubies, they also have either emeralds or diamonds (but not both simultaneously).
- If an adventurer has emeralds, they also have either rubies or sapphires (but not both simultaneously).

What is the minimum number of adventurers that can be in such a group?

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