
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 $f\left( x \right)=\sin (ωx+\dfrac{π}{3})$ $(ω > 0)$, the lines $x=m$, $x=n$ are any two symmetry axes of the graph of $y=f\left( x \right)$, and the minimum value of $\left| m-n \right|$ is $\dfrac{π}{4}$.

(1) Find the interval where $f\left( x \right)$ is increasing;

(2) After moving the graph of function $f\left( x \right)$ to the right by $\dfrac{π}{8}$ units, stretch the abscissa of each point on the obtained graph to twice its original length, keeping the ordinate unchanged, to obtain the graph of function $y={g}\left( x \right)$. If the equation ${g}\left( x \right)+k=0$ has two different real roots $α, β$ in the interval $\left[ 0,\dfrac{π}{2} \right]$, find $α+β$;

(3) Let $h\left( x \right)={{2}^{x}}-a$. For any ${{x}_{1}}\in \left[ 0,1 \right]$, there exists ${{x}_{2}}\in \left[ 0,\dfrac{5π}{24} \right]$ such that $h\left( {{x}_{1}} \right)=f({{x}_{2}})$, find the range of real number $a$.

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