
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
Let \( n \in \mathbf{Z}_{+} \). From the origin \( O \) along the coordinate grid \( (y=k, x=h, k, h \in \mathbf{Z}) \) to \( P(n, n) \), what is the total number of intersections with the line \( y = x \) for all shortest paths? (Including \((0,0)\) and \((n, n)\))?

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