
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
(1) Determine the range of the function $f(x)=\sin^2{x} + \cos{x} + 1$, where $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.  
(2) Find the domain and intervals of monotonicity for the function $y=\tan\left( \frac{x}{2} + \frac{\pi}{3}\right)$.

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