
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
A machine processes a positive integer $N$ according to the following rule:
- If $N$ is odd, output $O = 5N + 3$.
- If $N$ is even, output $O = \frac{N}{3}$ if $N$ is divisible by 3, otherwise $O = N + 1$.

Starting with $N = 6$, applying the machine rules five times consecutively leads to $1$. Using the same 5-step process, find the sum of all starting values $N$ such that the final output is $1$.
A) 6
B) 12
C) 18
D) 24
E) 30

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