
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
Define the second-order product sum of four numbers $a, b, c, d$ as $\begin{bmatrix} \overset{ab}{cd}\end{bmatrix}=ad+bc$. The third-order product sum of nine numbers can be calculated by converting it into a second-order product sum as follows: 
$$
\begin{bmatrix} 
a_{1}a_{2}a_{3} \\ 
b_{1}b_{2}b_{3} \\ 
c_{1}c_{2}c_{3}
\end{bmatrix}
= a_{1} \times \begin{bmatrix} \overset{b_{2}b_{3}}{c_{2}c_{3}}\end{bmatrix}
+ a_{2} \times \begin{bmatrix} \overset{b_{1}b_{3}}{c_{1}c_{3}}\end{bmatrix}
+ a_{3} \times \begin{bmatrix} \overset{b_{1}b_{2}}{c_{1}c_{2}}\end{bmatrix}.
$$
Given the function $f(n) = \begin{bmatrix} n & 2 & -9 \\ n & 1 & n \\ 1 & 2 & n\end{bmatrix}$ (where $n \in \mathbb{N}^+$), the minimum value of $f(n)$ is _______.

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