
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
Find the derivative of the following functions:

$(1) y = (3x^2 - 4x)(2x + 1);$

$(2) y = \sin\frac{x}{2}(1 - 2\cos^2\frac{x}{4});$

$(3) y = 3^x e^x - 2^x + e;$

$(4) y = \frac{\ln x}{x^2 + 1};$

$(5) y = \ln\frac{2x - 1}{2x + 1}.$

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