
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 each \( x \in \mathbb{R} \), let \( \{x\} \) be the fractional part of \( x \) in its decimal representation. For instance, \( \{3.4\} = 3.4 - 3 = 0.4 \), \( \{2\} = 0 \), and \( \{-2.7\} = -2.7 - (-3) = 0.3 \). Find the sum of all real numbers \( x \) for which \( \{x\} = \frac{1}{5} x \).

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