
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 functions $f(x)=x^{2}$, ($x\in[-2,2]$), and $g(x)=a^{2}\sin (2x+ \frac {π}{6})+3a$, ($x\in[0, \frac {π}{2}]$), for all $x_{1}\in[-2,2]$, there exists an $x_{0}\in[0, \frac {π}{2}]$ such that $g(x)=f(x_{1})$ holds true. Determine the range of values for the real number $a$.

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