
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 $f(x)=\sin (\omega x+\varphi)-b$ ($\omega > 0$, $0 < \varphi < \pi$), the distance between two adjacent axes of symmetry of its graph is $\dfrac{\pi}{2}$. If the graph of $f(x)$ is first translated to the right by $\dfrac{\pi}{6}$ units and then translated upwards by $\sqrt{3}$ units, the resulting function $g(x)$ is an odd function.  
(1) Find the analytical expression of $f(x)$.  
(2) Find the axes of symmetry and intervals of monotonicity of $f(x)$.  
(3) If for any $x \in \left[0, \dfrac{\pi}{3}\right]$, $f^{2}(x)-(2+m)f(x)+2+m \leqslant 0$ always holds, find the range of the real number $m$.

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