
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
For the elective course "4-5: Selected Topics on Inequalities", it is given that the function $f(x)= \frac{9}{{\sin }^{2}x}+ \frac{4}{{\cos }^{2}x},x\in\left(0, \frac{\pi}{2}\right)$, and $f(x)\geqslant t$ always holds.  
$(1)$ Find the maximum value of the real number $t$;  
$(2)$ When $t$ takes its maximum value, find the solution set of the inequality $\left|x+ \frac{t}{5}\right|+\left|2x-1\right|\leqslant 6$.

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