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

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

b) the fly will be at point \((8,10)\), having passed through the segment connecting the points (5,6) and (6,6) along the way;

c) the fly will be at point \((8,10)\), having passed within a circle of radius 3 with center at point \((4,5)\).

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