
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 a triangle $ABC$, let $a$, $b$, and $c$ denote the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively. Consider the function $f(x) = \sin(2x + B) + \sqrt{3}\cos(2x + B)$. It is known that $y = f\left(x - \frac{\pi}{3}\right)$ is an odd function.

(1) Find the interval(s) where $f(x)$ is strictly increasing.

(2) If $a = 1$ and $b = f(0)$, find the area $S$ of triangle $ABC$.

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