
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 \( n \geq 1 \) be an integer. How many ways can the rectangle having vertices \((0,0)\), \((n, 0)\), \((n, 1)\), \((0, 1)\) be dissected into \(2n\) triangles, all vertices of which have integer coordinates?

Remark: The triangles are considered as positioned on the coordinate plane; in particular, tilings related by rotation and reflection are considered distinct.

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