
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 \( S = \{1, 2, 3, 4\} \). Suppose \( a_1, a_2, \ldots, a_k \) is a sequence formed from the numbers in \( S \) such that it contains all permutations of \( S \) that do not end with 1. That is, for any permutation \( \left(b_1, b_2, b_3, b_4\right) \) of the 4 numbers in \( S \) where \( b_4 \neq 1 \), there exist indices \( i_1, i_2, i_3, i_4 \) such that
\[ a_{i_j} = b_j, \quad j = 1, 2, 3, 4 \text{ and } 1 \leq i_1 < i_2 < i_3 < i_4 \leq k. \]

Find the minimum value of \( k \) for the sequence.

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