
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
Jerry starts at 0 on the real number line and tosses a fair coin 10 times. When he gets heads, he moves one unit in the positive direction; when he gets tails, he moves one unit in the negative direction. However, once Jerry reaches or surpasses the position of 3 at any point, he cannot return to 0. Calculate the probability that Jerry reaches exactly 5 at any time during his journey and does not return to 0 at any point after reaching 3 in terms of reduced fraction $\frac{a}{b}$, where $a$ and $b$ are relatively prime integers. What is $a + b$?
A) 258
B) 262
C) 271
D) 289
E) 298

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