
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 \( a, b \in \mathbf{R} \), the function \( f(x) = a x - b \). If for any \( x \in [-1,1] \), \( 0 \leq f(x) \leq 1 \), find the range of \( \frac{3a + b + 1}{a + 2b - 2} \).

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