
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
Find all the real numbers  $x$  such that  $\frac{1}{[x]}+\frac{1}{[2x]}=\{x\}+\frac{1}{3}$  where  $[x]$  denotes the integer part of  $x$  and  $\{x\}=x-[x]$ .
For example,  $[2.5]=2, \{2.5\} = 0.5$  and  $[-1.7]= -2, \{-1.7\} = 0.3$ 

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