
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 $f(x)=\begin{cases} \frac{x}{3},0\leqslant x\leqslant \frac{1}{2} \
\frac{2{x}^{3}}{x+1}, \frac{1}{2} < x\leqslant 1\end{cases}$, and another function $g(x)=ax-\frac{a}{2}+3 (a>0)$. If for all $x_1\in[0,1]$, there exists an $x_2\in[0,\frac{1}{2}]$ such that $f(x_1)=g(x_2)$, determine the range of values for the real number $a$.

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