
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
Let \( f(x) = 4x^2 - 4(a+1)x + 3a + 3 \) where \( a \in \mathbf{R} \). If the equation \( f(x) = 0 \) has two distinct real roots both less than 2, determine whether the inequality \( (a+1)x^2 - ax + a - 1 < 0 \) holds for all real numbers \( x \).

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