
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 $f(x)= \frac {9}{\sin ^{2}x}+ \frac {4}{\cos ^{2}x}$, where $x\in(0, \frac {\pi}{2})$, and $f(x)\geqslant t$ always holds.  
$(1)$ Find the maximum value of the real number $t$;  
$(2)$ When $t$ takes the maximum value, find the solution set of the inequality $|x+ \frac {t}{5}|+|2x-1|\leqslant 6$.

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