
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
How many  $x$  are there such that  $x,[x],\{x\}$  are in harmonic progression (i.e, the reciprocals are in arithmetic progression)? (Here  $[x]$  is the largest integer less than equal to  $x$  and  $\{x\}=x-[ x]$  )

[list=1]
[*] 0
[*] 1
[*] 2
[*] 3
[/list]

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