
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 \( T_n \) with the grid lines' intersection points (a total of \( (n+1)^2 \) intersection points) as lattice points. The numbers \( 1, 2, \ldots, (n+1)^2 \) are assigned to all the lattice points of \( T_n \) such that different lattice points have different numbers. An \( 1 \times 1 \) cell \( S \) of \( T_n \) is called a "good cell" if the numbers on the four vertices of \( S \) read counterclockwise from any starting vertex are in increasing order. (For example, in a specific assignment of numbers \( 1, 2, \ldots, 9 \) to the lattice points of \( T_2 \), cells \( B \) and \( C \) are good cells, while \( A \) and \( D \) are not good cells.) Let \( f(n) \) denote the maximum number of good cells in \( T_n \).

(1) Find the value of \( f(2) \);
(2) Derive the expression for \( f(n) \) in terms of the positive integer \( n \).

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