
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 the value:  
(1) Calculate $$\int_{0}^{\frac{\pi}{2}}(\sin \frac{x}{2} + \cos \frac{x}{2})^2 dx$$;  
(2) Given a complex number $z$ satisfying $z \cdot \overline{z} - i(\overline{3z}) = 1 - (\overline{3i})$, find $z$.

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