
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
Divide a set $M$ containing $3n$ positive integers into three sets $A$, $B$, and $C$, each with equal numbers of elements and no common elements between them, where $A=\{a_1, a_2, \ldots, a_n\}$, $B=\{b_1, b_2, \ldots, b_n\}$, and $C=\{c_1, c_2, \ldots, c_n\}$. If the elements in $A$, $B$, and $C$ satisfy the conditions: $c_1 < c_2 < \ldots < c_n$ and $a_k + b_k = c_k$ for $k=1, 2, \ldots, n$, then $M$ is called a "complete union set".
1. If $M=\{1, x, 3, 4, 5, 6\}$ is a "complete union set", find the value of $x$.
2. For the "complete union set" $M=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$, among all sets $C$ that meet the conditions, find the set $C$ with the smallest product of elements.

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