
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
A fly starts at the origin on a coordinate grid and moves only to the right or up along the grid lines (monotone walk). At each grid point, the fly randomly chooses its next direction: up or right. Find the probability that at some point:

a) The fly will be at the point $(8, 10)$;

b) The fly will be at the point $(8, 10)$, passing through the segment connecting the points $(5, 6)$ and $(6, 6)$;

c) The fly will be at the point $(8, 10)$, passing inside a circle of radius 3 centered at the point $(4, 5)$.

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