
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 box contains a mixture of green and yellow playing cards, at most $2209$ in total. It is known that when three cards are selected randomly without replacement, there is a probability of exactly $\frac{1}{3}$ that all three are green or all three are yellow. What is the largest possible number of green cards in the box that is consistent with this data?

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