
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 $f\left(x\right)=A\sin \left(\omega x+ \dfrac{\pi}{6}\right)-1$ $(A > 0,\omega > 0)$, the distance between two adjacent axes of symmetry of its graph is $\dfrac{\pi}{2}$, and $f\left( \dfrac{\pi}{6}\right)=1$, find the maximum value of $f(x_{1})-f(x_{2})$ for any real numbers $x_{1}$ and $x_{2}$ in the interval $\left[0, \dfrac{\pi}{2} \right]$.

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