
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 $f(x) = 5\sin x\cos x - 5\sqrt{3}\cos^2 x + \frac{5\sqrt{3}}{2}$ ($x \in \mathbb{R}$).

(1) Find the period and the extremum values of $f(x)$;  
(2) Find the intervals where $f(x)$ is increasing;  
(3) Write the equation of the axis of symmetry and the coordinates of the center of symmetry for the graph of $f(x)$.

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