
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
Let the function $f(x) = \overrightarrow{a} \cdot (\overrightarrow{b} + \overrightarrow{c})$, where vector $\overrightarrow{a} = (\sin x, -\cos x)$, $\overrightarrow{b} = (\sin x, -3\cos x)$, $\overrightarrow{c} = (-\cos x, \sin x)$ ($a \in \mathbb{R}$).  
(1) Find the maximum value of the function $f(x)$ and its smallest positive period;  
(2) Find the intervals where $f(x)$ is increasing.

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