
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
Given a sequence $\{a_n\}$ where $a_n = n$, for each positive integer $k$, in between $a_k$ and $a_{k+1}$, insert $3^{k-1}$ twos (for example, between $a_1$ and $a_2$, insert three twos, between $a_2$ and $a_3$, insert $3^1$ twos, between $a_3$ and $a_4$, insert $3^2$ twos, etc.), to form a new sequence $\{d_n\}$. Let $S_n$ denote the sum of the first $n$ terms of the sequence $\{d_n\}$. Find the value of $S_{120}$.

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