
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
What is the necessary and sufficient condition for 

$$
\left\{ x^{2} \right\} > \left(\lfloor x \rfloor\right)^{2}
$$

where \( x \) is any positive number? (\(\lfloor a \rfloor\) denotes the greatest integer that is less than or equal to the number \( a \)).

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