
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 a real number \(\alpha\), let \( [\alpha] \) denote the greatest integer less than or equal to \(\alpha\). For example, \([3] = 3\), \([\sqrt{2}] = 1\), \([-\pi] = -4\). Suppose \(x\) is a positive real number. If \(\left[\log_{2} x\right]\) is even, then \(x\) is called a lucky number. What is the probability of randomly choosing a lucky number in the interval \((0, 1)\)?

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