
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 the expression \( S = \sqrt{x_{1} - x_{2} + x_{3} - x_{4}} \), \( x_{1}, x_{2}, x_{3}, x_{4} \) are a permutation of 1, 2, 3, and 4. Determine the number of distinct permutations that make \( S \) a real number.

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