
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
In a convex $n$-gon ($n \geq 4$) rose garden, each of the $n$ vertices has one red rose planted. There is a direct path connecting every pair of red roses, and these paths do not result in any three lines intersecting at a single point. These paths divide the garden into many non-overlapping regions (triangles, quadrilaterals, etc.), and each region contains one white rose (or black rose). Find the expression for the total number of roses $f(n)$ in the rose garden.

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