
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 $S = \{1, 2, 3, 4\}$. Let $a_{1}, a_{2}, \cdots, a_{k}$ be a sequence composed of numbers from $S$, which includes all permutations of $(1, 2, 3, 4)$ that do not end with 1. That is, if $\left(b_{1}, b_{2}, b_{3}, b_{4}\right)$ is a permutation of $(1, 2, 3, 4)$ and $b_{4} \neq 1$, then there exist indices $1 \leq i_{1} < i_{2} < i_{3} < i_{4} \leq k$ such that $\left(a_{i_{1}}, a_{i_{2}}, a_{i_{3}}, a_{i_{4}}\right)=\left(b_{1}, b_{2}, b_{3}, b_{4}\right)$. Find the minimum value of $k$.


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