
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
Let $S_n = \{1,2,3,\ldots,n\}$. If $X \subseteq S_n$, define the product of all elements in $X$ as the "capacity" of $X$ (if $X$ has only one element, then the capacity is the value of that element, and by convention the capacity of the empty set is 0). If the capacity of $X$ is an odd (even) number, then $X$ is called an odd (even) subset of $S_n$. Find the sum of the capacities of all odd subsets of $S_4$.

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