
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

For any region \( D \) on the Cartesian plane, let \( N(D) \) represent the number of lattice points in \( D \), and let \( n \) be a natural number greater than 1.

1. Let \( A \) be the region enclosed by the curve \( y = x^2 \) and the lines \( x = n \) and \( y = 1 \) (including the boundary). Find \( N(A) \).

2. Let \( B \) be the region enclosed by the curve \( y = x^2 \) and the lines \( x = 1 \) and \( y = n^2 \) in the first quadrant (including the boundary). Find \( N(B) \).

3. Compute \( \sum_{k=1}^{n^2} \lfloor \sqrt{k} \rfloor \).

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