
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 $\overrightarrow{a}=(\cos x,\sin x)$ and $\overrightarrow{b}=(\cos x, \sqrt {3}\cos x)$, let $f(x)=2 \overrightarrow{a}\cdot \overrightarrow{b}+1$.  
$(1)$ Find the range of $f(x)$ when $x\in[0, \frac {\pi}{2}]$;  
$(2)$ If for any $x\in[0, \frac {\pi}{2}]$ and any $\alpha\in[ \frac {\pi}{12}, \frac {\pi}{3}]$, the inequality $k\cdot \sqrt {1+\sin 2\alpha}-\sin 2\alpha\leqslant f(x)+1$ always holds, find the range of the real number $k$.

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