
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
A large candle is $150$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $12$ seconds to burn down the first centimeter from the top, $24$ seconds to burn down the second centimeter, and $12k$ seconds to burn down the $k$-th centimeter. Suppose it takes \( S \) seconds for the candle to burn down completely. Compute the candle's height in centimeters \( \frac{S}{2} \) seconds after it is lit.

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