
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 that \(x\) is a non-negative real number and \(i=1,2,3,4\), with the equation \(x_{1} + x_{2} - x_{3} + x_{4} = 1\), define \(S = 1 - \sum_{t=1}^{4} x_{t}^{4} - 6 \sum_{1 \leq j < k \leq 4} x_{j} x_{k}\). Find the range of possible values for \(S\).

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