
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 vectors $\overrightarrow{a}=(\cos^2x-\sin^2x,\frac{1}{2}),\overrightarrow{b}=(\frac{1}{2},\sin^2x+\sqrt{3}\sin x \cos x)$, and a function $f(x)=\overrightarrow{a}•\overrightarrow{b}$.  
$(1)$ Find the axis and center of symmetry of the graph of the function $f(x)$.  
$(2)$ Change all the abscissas of the points on the graph of the function $f(x)$ to $\frac{1}{2}$ (keeping the ordinates unchanged), then shift the resulting graph to the left by $\frac{π}{6}$ units to obtain the graph of the function $y=g(x)$. If the inequality $6a^{2}-a-\frac{5}{4}⩾g(x)$ holds for all $x$ in $[-\frac{π}{4},\frac{π}{8}]$, find the range of real numbers for $a$.

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