
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
On a board, the 2014 positive integers from 1 to 2014 are written. The allowed operation is to choose two numbers \( a \) and \( b \), erase them, and write in their place the numbers \( \text{gcd}(a, b) \) (greatest common divisor) and \( \text{lcm}(a, b) \) (least common multiple). This operation can be performed with any two numbers on the board, including numbers that resulted from previous operations. Determine the largest number of 1's that we can leave on the board.

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