
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
Jacob begins a sequence with the first term being 10. To generate each succeeding term, he flips a fair coin. If it lands on heads, he doubles the previous term and subtracts 1. If it lands on tails, he halves the previous term and subtracts 2. Additionally, if at any step the resultant number is divisible by 3, he must add 1 to that result. What is the probability that the fourth term in Jacob's sequence is an integer?
A) $\frac{1}{4}$
B) $\frac{1}{3}$
C) $\frac{1}{2}$
D) $\frac{2}{3}$
E) $\frac{3}{4}$

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