
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
The numbers $a_1, a_2, a_3, a_4, b_1, b_2, b_3, b_4, c_1, c_2, c_3, c_4, d_1, d_2, d_3, d_4$ are equal to the numbers $1, 2, 3, ..., 12$ in some order. Find the smallest possible value of
\[
S = a_1 a_2 a_3 a_4 + b_1 b_2 b_3 b_4 + c_1 c_2 c_3 c_4 + d_1 d_2 d_3 d_4.
\]

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