
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 vectors $\overrightarrow{a} = (1+\sin 2x, \sin x - \cos x)$ and $\overrightarrow{b} = (1, \sin x + \cos x)$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$:
1. Find the smallest positive period of the function $f(x)$.
2. Find the maximum value of $f(x)$ and the set of $x$ values for which this maximum is achieved.

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