
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 basket contains a mixture of black pens and red pens, at most $2500$ in total. It is known that, when two pens are selected randomly without replacement, there is a probability of exactly $\frac{1}{3}$ that both are black or both are red. What is the largest possible number of black pens in the basket that is consistent with this data?

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