
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
a) Given four distinct real numbers \( a_{1} < a_{2} < a_{3} < a_{4} \), arrange these numbers in such an order \( a_{i_{1}}, a_{i_{2}}, a_{i_{3}}, a_{i_{4}} \) (where \( i_{1}, i_{2}, i_{3}, i_{4} \) are the numbers 1, 2, 3, 4 in some permutation) so that the sum

\[
\Phi = (a_{i_{1}} - a_{i_{2}})^{2} + (a_{i_{2}} - a_{i_{3}})^{2} + (a_{i_{3}} - a_{i_{4}})^{2} + (a_{i_{4}} - a_{i_{1}})^{2}
\]

is as small as possible.

b) Given \( n \) distinct real numbers \( a_{1}, a_{2}, a_{3}, \ldots, a_{n} \), arrange these numbers in such an order \( a_{i_{1}}, a_{i_{2}}, a_{i_{3}}, \ldots, a_{i_{n}} \) so that the sum

\[
\Phi = (a_{i_{1}} - a_{i_{2}})^{2} + (a_{i_{2}} - a_{i_{3}})^{2} + \ldots + (a_{i_{n-1}} - a_{i_{n}})^{2} + (a_{i_{n}} - a_{i_{1}})^{2}
\]

is as small as possible.

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