
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 a function $y=f(x)$ with a domain of $\mathbb{R}$, for any $x \in \mathbb{R}$, it always holds that $f(2x)=-2f(x)$.
(1) If $f(1)=-3$, find the value of $f(16)$;
(2) If $f(x)=x^{2}-2x+2$ for $x \in (1,2]$, find the explicit expression and range of $y=f(x)$ for $x \in (1,8]$;
(3) If $f(x)=-|x- \dfrac {3}{2}|$ for $x \in (1,2]$, find the maximum and minimum values of $y=f(x)$ in the interval $(1,2^{n}]$ for $n \in \mathbb{N^*}$.

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