
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 $$f(\log_{2}x)=ax^{2}-2x+1-a$$, where $a\in\mathbb{R}$.
1. Find the expression for $f(x)$.
2. Solve the equation $f(x)=(a-1)\cdot4^{x}$ with respect to $x$.
3. Let $h(x)=2^{-x}f(x)$, if $a\geq \frac{1}{2}$, then for any $x_{1}, x_{2}\in[-1,1]$, it always holds that $$|h(x_{1})-h(x_{2})|\leq \frac{a+1}{2}$$. Find the range of $a$.

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