
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  $n$  be positive numbers. Consider the following region on  $xy$  plane.

\[ D_n=\left\{(x,y) |\ \frac{x}{n+\frac{1}{2}}\leq y\leq [x+1]-x,\ x\geq 0 \right\}. \]

Here  $\ [x]$  is the greatest integer  $\ \leq x.$  Find the area of  $D_n.$ 

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