
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 gigantic candle is $150$ centimeters tall. The candle burns down at a varying rate determined by the formula $10k^2$ seconds for the $k$-th centimeter from the top. Calculate the total time $T$ it takes for the candle to burn down completely. Additionally, find the height of the candle in centimeters $\frac{T}{2}$ seconds after it is lit.

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