
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 function $f(x)=\overrightarrow{m}\cdot \overrightarrow{n}$, where $\overrightarrow{m}=(\sin x+\cos x, \sqrt{3}\cos x)$ and $\overrightarrow{n}=(\cos x-\sin x, 2\sin x)$. In acute triangle $\triangle ABC$, with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. <br/>$(1)$ If $f(A)=1$, find the value of angle $A$; <br/>$(2)$ Under the condition in $(1)$, if $a=\sqrt{3}$, find the range of values for $c+b$.

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