
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
Find the number of paths from the point \((0, 0)\) to the point \((10, 5)\) that do not pass through the segments \(AB, CD, EF, GH\). Given the coordinates \(A(2,2), B(3,2), C(4,2), D(5,2), E(6,2), F(6,3), G(7,2), H(7,3)\), and the total number of paths from \((0, 0)\) to \((10, 5)\) is \(\binom{15}{5} = 3003\).

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