
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

A vertical system of tubes with five levels is given, illustrated in the diagram. At each subsequent level, the tube splits into two. A ball is dropped from the top. The tubes are arranged such that the ball moves to the left with a probability of $1 / 3$ and to the right with a probability of $2 / 3$. Find the probability that the ball reaches the bottom tube. Provide the answer to three decimal places.

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