
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 every real number \( x \), let \( \lfloor x \rfloor \) denote the greatest integer less than or equal to \( x \). The fractional part of \( x \), denoted as \( \langle x \rangle \), is defined as \( \langle x \rangle = x - \lfloor x \rfloor \). How many real numbers \( x \) are there such that \( 1 \leqslant x \leqslant 10 \) and \( \langle x \rangle^{2} = \langle x^{2} \rangle \)?

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