
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
Two numbers are written on a board in the laboratory. Each day, the senior researcher Petya erases both numbers from the board and writes their arithmetic mean and harmonic mean instead. On the morning of the first day, the numbers written on the board are 1 and 2. Find the product of the numbers written on the board in the evening of the 1999th day. (The arithmetic mean of two numbers \( a \) and \( b \) is given by \( \frac{a+b}{2} \), and the harmonic mean is given by \( \frac{2}{\frac{1}{a} + \frac{1}{b}} \).)

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