
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 how many positive integers \( n \leq 1000 \) does the equation in real numbers
\[ x^{\lfloor x \rfloor} = n \]
have a positive solution for \( x \)? (For a real number \( x \), \(\lfloor x \rfloor\) denotes the largest integer that is not greater than \( x \)).

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