
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 integer part of  $x$ ;  $\{ x \}$  be the fractional part of  $x$  ( $\{x\}$   $=$   $x-$   $\lfloor x \rfloor$ ). Let  $A$  denote the set of all real numbers  $x$  satisfying   $$ \{x\} =\frac{x+\lfloor x \rfloor +\lfloor x + (1/2) \rfloor }{20} $$  If  $S$  is the sume of all numbers in  $A$ , find  $\lfloor S \rfloor$ 

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