
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 the function \( f(x) = \frac{2}{3} x^{3} + \frac{1}{2}(a-1) x^{2} + a x + 1 \), determine the range of real number \( a \) such that \( f(x) \) is an increasing function on the interval \( (1, 2) \).

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