
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
In the Cartesian coordinate system $xoy$, there is a circle $O: x^{2}+y^{2}=1$, and another circle $M: (x+a+1)^{2}+(y-2a)^{2}=1$ (where $a$ is a real number). If there exist points $P$ on circle $O$ and $Q$ on circle $M$ such that $\angle OQP=30^{\circ}$, then the range of values for $a$ is \_\_\_\_\_\_.

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