
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 $f(x)=4\sin\frac{x}{2}\cos(\frac{x}{2}+\frac{π}{3})+\sqrt{3}$.  
$(1)$ If $f(α)=\frac{1}{3}$, find $cos(α+\frac{5}{6}π)$;  
$(2)$ If the inequality $|f^{2}(x)-\lambda |\leqslant f\left(x\right)+2$ holds for all $x∈[-\frac{π}{6},\frac{π}{3}]$, find the range of values for $\lambda$.

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