
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
Given the graph and conditions below, where \( n < m \) are positive integers, and a particle starts at the origin \( A(0,0) \) and moves to the endpoint \( C(m, n) \) with non-decreasing moves. Let point \( B(n, n) \) be a reference point.

(i) Determine the number of paths that lie entirely below the line \( AB \) and touch the line at most once.

(ii) Determine the number of paths that do not intersect the line \( AB \).

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