
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 functions $f\left(x\right)=\sin (ωx+\frac{π}{4})$ and $g\left(x\right)=2\sin (\frac{4}{3}x-\frac{π}{3})-1$, and it satisfies $\forall x\in \left[0,\pi \right]$, $f\left(x\right)\cdot g\left(x\right)\leqslant 0$ always holds.<br/>$(1)$ Find the zeros of $g\left(x\right)$ and the analytical expression of $f\left(x\right)$.<br/>$(2)$ Find the range of the difference between the maximum and minimum values of the function $f\left(x\right)$ on the interval $[t,t+\frac{π}{4}]$.

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