
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 $\overrightarrow{a}=(m,1)$, $\overrightarrow{b}=(\sin x,\cos x)$, and $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$ satisfies $f\left( \frac {\pi}{2}\right)=1$.  
$(1)$ Find the analytical expression of the function $y=f(x)$;  
$(2)$ Find the minimum positive period, the extremum values of the function $y=f(x)$, and the corresponding $x$ values;  
$(3)$ In an acute-angled $\triangle ABC$, if $f\left( \frac {\pi}{12}\right)= \sqrt {2}\sin A$ and $AB=2$, $AC=3$, find the length of $BC$.

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