
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
Suppose  $x$  is a positive real number such that  $\{x\}, [x]$  and  $x$  are in a geometric progression. Find the least positive integer  $n$  such that  $x^n > 100$ . (Here  $[x]$  denotes the integer part of  $x$  and  $\{x\} = x - [x]$ .)

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