
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 is crawling from the origin along the lines of an integer grid, moving only to the right or up (monotonic wandering). At each grid node, the fly randomly chooses its next direction: upward or to the right. Find the probability that at some moment:

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 $(6, 6)$ along the way;

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

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