
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 candle is 100 centimeters tall. The time taken to burn down each centimeter from the top increases quadratically. Specifically, it takes $5k^2$ seconds to burn down the $k$-th centimeter. Determine the height of the candle in centimeters $\tfrac{T}{2}$ seconds after it is first lit, where $T$ is the total time taken for the candle to burn down completely.

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