
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
Let $a$ be a positive number, and the function $f(x)=ax^{2}+bx+c$ satisfies $f(0)=1$ and $f(x)=f(\frac{2}{a}-x)$.<br/>$(1)$ If $f(1)=1$, find $f(x)$;<br/>$(2)$ Let $g(x)=\log _{2}(x-2\sqrt{x}+2)$, if for any real number $t$, there always exist $x_{1}$, $x_{2}\in [t-1,t+1]$, such that $f(x_{1})-f(x_{2})\geqslant g(x_{3})-g(x_{4})$ holds for all $x_{3}$, $x_{4}\in [\frac{1}{4},4]$, find the range of values for $a$.

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