
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 $S = \{(x, y) \mid -2 \leq y \leq |x|, -2 \leq x \leq 2\}$, determine the probability that for $(x, y) \in S$, one root of the quadratic equation $t^{2} + (|x| - 1)t + |y| - 2 = 0$ is greater than 1 and the other root is less than 1.

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