
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
Define the function  $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$  for all positive real numbers  $x$ . How many integers from  $1$  to  $2023$  inclusive are in the range of  $f(x)$ ? Note that  $\lfloor x\rfloor$  is known as the  $\textit{floor}$  function, which returns the greatest integer less than or equal to  $x$ .

*Proposed by Harry Kim*

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