
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
At a round rotating table with 8 white and 7 black cups seated are 15 gnomes. The gnomes are wearing 8 white and 7 black hats. Each gnome takes a cup that matches the color of their hat and places it in front of them. After this, the table rotates randomly. What is the maximum number of color matches between the cups and hats that can be guaranteed after the table rotates (the gnomes can choose how to sit, but do not know how the table will rotate)?

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