
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
Decomposing the positive integer $12$ into the product of two positive integers can be done in three ways: $1 \times 12$, $2 \times 6$, and $3 \times 4$. Among these, $3 \times 4$ has the smallest absolute difference between the two numbers, and we call $3 \times 4$ the optimal decomposition of $12$. When $p \times q$ ($p \leqslant q$ and $p, q \in \mathbb{N}^*$) is the optimal decomposition of a positive integer $n$, we define the function $f(n) = q - p$. For example, $f(12) = 4 - 3 = 1$. Then, $f(81) = $ \_\_\_\_\_\_ , and the sum of the first $100$ terms of the sequence $\{f(3^{n})\} (n \in \mathbb{N}^*)$ is \_\_\_\_\_\_ .

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