
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
Consider the sequence \(1, 3, 4, 9, 10, 12, 13, \ldots\) consisting of integers greater than or equal to 1, in increasing order, that are either powers of 3 or sums of distinct powers of 3 (for example: \(4=3^{1}+3^{0}, 10=3^{2}+3^{0}, 13=3^{2}+3^{1}+3^{0}\)). 

Which integer is in the hundredth position?

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