
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 the sequence ${{a_n}}$, where $a_n = \begin{cases} 2^{n-1} & \text{for positive odd } n \\ 2n-1 & \text{for positive even } n \end{cases}$, let $S_n$ denote the sum of the first $n$ terms of the sequence. Find $S_{12}$. Please provide a numeric answer.

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