
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 $f(x) = 2\cos\left(\frac{\pi}{2} - \omega x\right) + 2\sin\left(\frac{\pi}{3} - \omega x\right)$ ($\omega > 0$, $x \in \mathbb{R}$), if $f\left(\frac{\pi}{6}\right) + f\left(\frac{\pi}{2}\right) = 0$, and $f(x)$ is decreasing in the interval $\left(\frac{\pi}{6}, \frac{\pi}{2}\right)$.

(1) Find the value of $f(0)$;  
(2) Find $\omega$;  
(3) Solve the inequality $f(x) \geq 1$.

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