
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
Natalie has a copy of the unit interval \([0,1]\) that is colored white. She also has a black marker, and she colors the interval in the following manner: at each step, she selects a value \(x \in [0,1]\) uniformly at random, and

(a) If \(x \leq \frac{1}{2}\) she colors the interval \(\left[x, x+\frac{1}{2}\right]\) with her marker.
(b) If \(x > \frac{1}{2}\) she colors the intervals \([x, 1]\) and \(\left[0, x-\frac{1}{2}\right]\) with her marker.

What is the expected value of the number of steps Natalie will need to color the entire interval black?

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