
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 a binary number consisting of $n$ ones and $n$ zeros, forming a $2n$-bit number, find the count of such numbers where the cumulative count of ones is always greater than the cumulative count of zeros when scanning from left to right up to the $(2n-1)$-th position. This problem can be transformed to the problem of counting the number of paths from point $O$ to point $A$ that stay above the diagonal $OA$. Equivalently, it is the problem of counting the paths from point $O^{\prime}(0,1)$ that do not touch the diagonal $OA$ and reach point $A^{\prime}(n-1, n)$. This converts it to the problem in the previous example.

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