
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 on an integer grid. The fly moves only to the right or upward along the lines of the grid (monotonic wandering). At each grid node, the fly randomly chooses the direction of further movement: up or 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 \((8, 10)\), passing through the segment connecting the points \((5, 6)\) and \((6, 6)\) along the way;

c) the fly will be at the point \((8, 10)\), passing inside a circle of radius 3 centered at the point \((4, 5)\).

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