
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
Sebestyén receives a prismatic cake with a triangular base on each birthday starting from his third birthday. Each cake has candles at the three vertices of the upper triangular base, and additional candles on the top such that the total number of candles equals his age, ensuring that no three candles are collinear. Sebestyén wants to cut the cake into triangular slices, each slice having its vertices at the locations of the candles (with no candles inside the slices). How many slices can he cut the cake into on his $k$-th birthday?

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