
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 crawls from the origin on an integer grid, moving only in a monotonic fashion (either to the right or upward). At each lattice point, the fly randomly decides whether to move up or right. Find the probability that at some point:

a) the fly reaches the point (8, 10);

b) the fly reaches the point (8, 10) after passing through the segment connecting the points (5, 6) and (6, 6);

c) the fly reaches the point (8, 10) after passing within a circle of radius 3 centered at the point (4, 5).

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