
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
Given the sets \( A = \{a_1, a_2, a_3, a_4\} \) and \( B = \{a_1^2, a_2^2, a_3^2, a_4^2\} \), where \( a_1 < a_2 < a_3 < a_4 \) and \( a_1, a_2, a_3, a_4 \in \mathbf{N}^{\top} \). If \( A \cap B = \{a_1, a_4\} \), \( a_1 + a_4 = 10 \), and the sum of all elements in \( A \cup B \) is 124, find the sets \( A \) and \( B \).

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