
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 $\overrightarrow{a}=(\cos \theta,\sin \theta)$, $\overrightarrow{b}=(\cos \phi,\sin \phi)$  
$(1)$ If $|\theta-\phi|= \dfrac {\pi}{3}$, find the value of $|\overrightarrow{a}- \overrightarrow{b}|$;  
$(2)$ If $\theta+\phi= \dfrac {\pi}{3}$, let $f(\theta)= \overrightarrow{a}\cdot \overrightarrow{b}-\lambda|\overrightarrow{a}+ \overrightarrow{b}|$, where $\theta\in[0, \dfrac {\pi}{2}]$. When $1\leqslant \lambda\leqslant 2$, find the minimum value of $f(\theta)$.

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