
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 moving from the origin on a grid. The fly can only move along the lines of the integer grid to the right or upwards (monotonic walk). 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 end up at the point \((8, 10)\);

b) The fly will end up at the point \((8, 10)\), passing through the segment connecting the points \((5, 6)\) and \((6, 6)\) on the way;

c) The fly will end up at the point \((8, 10)\), passing through the interior of a circle with radius 3 centered at the point \((4, 5)\).

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