
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 $f(x)=ax^{2}+\\ln x(a∈R)$.
(1) When $a= \\dfrac {1}{2}$, find the maximum and minimum values of $f(x)$ in the interval $[1,e]$;
(2) If function $g(x)$, $f\_{1}(x)$, $f\_{2}(x)$, are defined in the common domain $D$, and satisfy $f\_{1}(x) < g(x) < f\_{2}(x)$, then $g(x)$ is called the "active function" of $f\_{1}(x)$, $f\_{2}(x)$. Given functions $f\_{1}(x)=(a- \\dfrac {1}{2})x^{2}+2ax+(1-a^{2})\\ln x,f\_{2}(x)= \\dfrac {1}{2}x^{2}+2ax$. If in the interval $(1,+∞)$, function $f(x)$ is the "active function" of $f\_{1}(x)$, $f\_{2}(x)$, find the range of $a$.

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