
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 set \( S = \{1, 2, 3, 4\} \), let \( a_{1}, a_{2}, \cdots, a_{k} \) be a sequence composed of numbers from \( S \), containing all permutations of \( (1, 2, 3, 4) \) that do not end with 1. That is, if \( (b_{1}, b_{2}, b_{3}, b_{4}) \) is a permutation of \( (1, 2, 3, 4) \) with \( b_{4} \neq 1 \), then there exist indices \( 1 \leqslant i_{1} < i_{2} < i_{3} < i_{4} \leqslant k \) such that
$$
(a_{i_{1}}, a_{i_{2}}, a_{i_{3}}, a_{i_{4}}) = (b_{1}, b_{2}, b_{3}, b_{4}).
$$

Find the minimum value of \( k \), the length of such a sequence.

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