
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 $f(x) = 2\cos^2{x} + \sin\left(2x - \frac{\pi}{6}\right)$:
1. Find the interval(s) where $f(x)$ is monotonically increasing; its maximum value, and the set of $x$ values for which it reaches this maximum value.
2. In triangle $ABC$, the sides opposite angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $f(A) = \frac{3}{2}$ and $b+c=2$, find the range of the real number $a$.

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