
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 $\overrightarrow{a}=(2\cos x,2\sin x)$, $\overrightarrow{b}=(\sin (x-\frac{π}{6}),\cos (x-\frac{π}{6}))$, and the function $f(x)=\cos \langle \overrightarrow{a},\overrightarrow{b} \rangle$.  
$(1)$ Find the monotonic intervals of the function $y=f(x)$.  
$(2)$ If in an acute triangle $\triangle ABC$, the sides opposite to the angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $f(A)=1$, find the range of $\frac{b+c}{a}$.

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