
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 vectors $\overrightarrow {a}=(\cos \frac {3}{2}x，\sin \frac {3}{2}x)$ and $\overrightarrow {b}=(\cos \frac {1}{2}x，\sin \frac {1}{2}x)$, where $x\in[0，\pi]$:
1. When $x= \frac {\pi}{4}$, find the value of $\overrightarrow {a}\cdot \overrightarrow {b}$ and $| \overrightarrow {a}+ \overrightarrow {b}|$.
2. Find the maximum value of $f(x)=m| \overrightarrow {a}+ \overrightarrow {b}|- \overrightarrow {a}\cdot \overrightarrow {b}$ ($m\in\mathbb{R}$).

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