
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 real number $x \in \left(0, \frac{\pi}{2}\right)$, let $f(x) = \frac{1}{9\sin^{2}x} + \frac{4}{9\cos^{2}x}$.

(1) If $f(x) \geq t$ always holds, find the maximum value $M$ of $t$;  
(2) Under the condition of (1), find the solution set of the inequality $x^2 + |x-2| + M \geq 3$.

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