
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
The distance between two adjacent symmetric axes of the function $f(x)=\cos(\omega x + \phi)$ $(\omega > 0, |\phi| < \frac{\pi}{2})$ is $\frac{\pi}{2}$, and $f(\frac{\pi}{6})=1$. <br/>$(1)$ Find the interval where $f(x)$ is monotonically decreasing;<br/>$(2)$ When $x \in [-\frac{\pi}{6}, \frac{\pi}{3}]$, the equation $f(x) - a = 0$ has a solution. Find the range of real numbers for $a$.

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