
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 $\overrightarrow {m}=(2a\cos x，\sin x)$, $\overrightarrow {n}=(\cos x，b\cos x)$, and the function $f(x)= \overrightarrow {m}\cdot \overrightarrow {n}- \frac { \sqrt {3}}{2}$, the y-intercept of function $f(x)$ is $\frac { \sqrt {3}}{2}$, and the coordinates of the highest point closest to the y-axis are $( \frac {π}{12}，1)$.  
(Ⅰ) Find the analytical expression of function $f(x)$;  
(Ⅱ) Shift the graph of function $f(x)$ to the left by $\varphi$ ($\varphi＞0$) units, and then stretch the x-coordinates of each point on the graph to twice their original length, obtaining the graph of the function $y=\sin x$. Find the minimum value of $\varphi$.

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