
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. He 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. What is the probability that he reaches or exceeds $6$ units in the positive direction at some point during the $10$ tosses, expressed as a reduced fraction $\frac{a}{b}$, where $a$ and $b$ are coprime integers?
A) $\frac{386}{1024}$
B) $\frac{193}{512}$
C) $\frac{210}{1024}$
D) $\frac{120}{512}$

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