
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 drawer contains a mixture of red socks and blue socks, at most $2000$ in all. Initially, the drawer has equal numbers of red and blue socks. One day, a number of blue socks were lost, making the total number uneven. After this event, it is known that when two socks are selected randomly without replacement, there is a probability of exactly $\frac{1}{2}$ that both socks are red or both are blue. What is the largest possible number of red socks in the drawer that corresponds with this new set-up?

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