
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 $f(x)=\sin 2x+2\cos ^{2}x+1$, $x∈[{0,\frac{π}{2}}]$.  
$(1)$ Find the range of the function $y=f(x)$;  
$(2)$ Find the interval where the function $y=f(x)$ is strictly increasing;  
$(3)$ If the inequality $a\cdot f(x)+2a\geqslant f(x)$ holds for all $x∈[0,\frac{π}{2}]$, find the range of real number $a$.

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