
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
Starting with the number 0, Casey performs an infinite sequence of moves as follows: he chooses a number from \(\{1, 2\}\) at random (each with probability \(\frac{1}{2}\)) and adds it to the current number. Let \(p_{m}\) be the probability that Casey ever reaches the number \(m\). Find \(p_{20} - p_{15}\).

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