
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 a real number $x \in \left(0, \frac{\pi}{2}\right)$, let $f(x)= \frac{1}{9\sin^2x}+ \frac{4}{9\cos^2x}$.  
$(I)$ If $f(x) \geqslant t$ always holds, find the maximum value of $t$;  
$(II)$ Under the condition of $(I)$, find the solution set of the inequality $|x+t|+|x-2| \geqslant 5$.

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