
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  $119$  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  $10$  seconds to burn down the first centimeter from the top,  $20$  seconds to burn down the second centimeter, and  $10k$  seconds to burn down the  $k$ -th centimeter.  Suppose it takes  $T$  seconds for the candle to burn down completely.  Then  $\tfrac{T}{2}$  seconds after it is lit, the candle's height in centimeters will be  $h$ .  Find  $10h$ .

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