
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
Calculate:  
(1) $$\int_{1}^{2} \left( \frac{1}{\sqrt{x}} + \frac{1}{x} + \frac{1}{x^2} \right) dx$$  
(2) Given complex numbers $z_1 = a + 2i$ ($a \in \mathbb{R}$), $z_2 = 3 - 4i$, and $\frac{z_1}{z_2}$ is a pure imaginary number, find $|z_1|$.

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