
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 $\overrightarrow{a}=(\sin \frac{\pi}{2}x,\cos \frac{\pi}{2}x)$, $\overrightarrow{b}=(\sin \frac{\pi}{2}x, \sqrt{3}\sin \frac{\pi}{2}x)$, $x\in R$, and the function $f(x)= \overrightarrow{a}\cdot ( \overrightarrow{a}+2 \overrightarrow{b})$, find:
(1) the smallest positive period of $f(x)$;
(2) the maximum and minimum values of $f(x)$ on the interval $[0,1]$, as well as the values of $x$ when the maximum and minimum values are obtained.

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