
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 a positive integer. Consider an \( n \times n \) grid \( T_n \), where the intersection points of the grid lines (a total of \((n+1)^2\) intersection points) are called lattice points. We assign the numbers \( 1, 2, \ldots, (n+1)^2 \) to all the lattice points of \( T_n \) such that different lattice points receive different numbers.

A \( 1 \times 1 \) subgrid \( S \) of \( T_n \) is called a "good subgrid" if the numbers read counterclockwise from one of its vertices increase sequentially. Let \( f(n) \) denote the maximum number of good subgrids in \( T_n \).

(1) Find the value of \( f(2) \).

(2) Determine the expression for \( f(n) \) in terms of the positive integer \( n \).

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