
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 begins at $0$ on the real number line and tosses a fair coin $10$ times. When he gets heads, he moves $1$ unit in the positive direction; when he gets tails, he moves $1$ unit in the negative direction. Calculate the probability that Jerry reaches exactly $3$ at any time during these tosses. Express your answer as a fraction $\frac{a}{b},$ where $a$ and $b$ are relatively prime positive integers, and find $a + b.$
A) $\frac{15}{64}$
B) $\frac{15}{128}$
C) $\frac{3}{128}$
D) $\frac{1}{64}$
E) $\frac{7}{32}$

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