
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
During a measurement, it was found that a set of data $(x\_i, y\_i)$ has a strong correlation, and the calculated regression line is $\hat{y} = x + 1$. Among these data points, $(1, y\_0)$ was not clearly written, with only the information that $y\_0$ is any value in $[0, 3]$. What is the probability that the absolute value of the residual corresponding to this data is not greater than $1$? (Note: Residual = Actual value - Predicted value)

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