
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
If \( A \) is a set of positive integers, take the set \( A' \) to be all elements which can be written as \( \pm a_1 \pm a_2 \ldots \pm a_n \), where \( a_i \) are distinct elements of \( A \). Similarly, form \( A" \) from \( A' \). What is the smallest set \( A \) such that \( A" \) contains all of \( 1, 2, 3, \ldots, 40 \)?

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