
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
Using $[x]$ to denote the greatest integer less than or equal to the real number $x$. For example, $[3]=3$, $[\sqrt{2}]=1$, $[-\pi]=-4$. Let $x$ be a positive real number. If $\left[\log _{2} x\right]$ is an even number, then $x$ is called a lucky number. What is the probability of randomly selecting a lucky number in the interval $(0,1)$?

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