
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

In a $3 \times 4$ rectangle, the natural numbers $1, 2, 3, \ldots, 12$ were written, each occurring exactly once. The table had the property that in each column, the sum of the top two numbers is twice the bottom number. Over time, some numbers were erased. Find all possible numbers that could have been written in the place of $\star$.

| 6 |  |  |  |
| :--- | :--- | :--- | :--- |
| 4 |  |  |  |
|  | 8 | 11 | $\star$ |

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