
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 moves from the origin of the coordinate grid. The fly only moves along the integer grid lines to the right or upward (monotone wandering). At each grid node, the fly randomly chooses the direction of further movement: upward or to the right. Find the probability that at some point:

a) the fly ends up at the point $(8,10)$;

b) the fly ends up at the point $(8,10)$ after passing through the segment connecting the points $(5,6)$ and $(6,6)$;

c) the fly ends up at the point $(8,10)$ after passing inside the circle of radius 3 centered at the point $(4,5)$.

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