
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 and can only move to the right or up along the lines of an integer grid (monotone wandering). At each point in the grid, the fly chooses randomly to move either up or to the 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 \((8, 10)\), having passed 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 through the interior of a circle of radius 3 centered at the point \((4, 5)\).

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