
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
The number \( x \) is chosen randomly from the interval \( (0,1] \). Define \( y = \left\lceil \log_{4} x \right\rceil \). Find the sum of the lengths of all subintervals of \( (0,1] \) for which \( y \) is odd. For any real number \( a, \left\lceil a \right\rceil \) is defined as the smallest integer not less than \( a \).

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