
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 $f(x) = 2\sin^2x + \sin x \cdot \cos x + \cos^2x$, where $x \in \mathbb{R}$. Find:  
(1) The value of $f\left( \frac{\pi}{12} \right)$;  
(2) The minimum value of the function $f(x)$ and the corresponding value of $x$;  
(3) The increasing interval of the function $f(x)$.

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