
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
For any real number \( x \), let \( \lfloor x \rfloor \) denote the floor of \( x \), which is the greatest integer less than or equal to \( x \). Let \( \{ x \} \) denote the fractional part of \( x \), which is \( \{ x \} = x - \lfloor x \rfloor \). 

For example, \( \lfloor 3.1 \rfloor = 3 \) and \( \{ 3.1 \} = 3.1 - 3 = 0.1 \). Similarly, \( \lfloor -2.7 \rfloor = -3 \) and \( \{ -2.7 \} = -2.7 - (-3) = 0.3 \).

Find all real numbers \( x \) such that \( \lfloor x \rfloor \times \{ x \} = 2019 \times x \).

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