
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
The numbers \( x_{1} < x_{2} < x_{3} < x_{4} \) are given. For which permutation of the indices \( 1, 2, 3, 4 \) as \( a, b, c, d \) will the following sum be minimized?

$$
\left(x_{a}-x_{b}\right)^{2}+\left(x_{b}-x_{c}\right)^{2}+\left(x_{c}-x_{d}\right)^{2}+\left(x_{d}-x_{a}\right)^{2}
$$

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