
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 $f(x)=ax^{2}+bx+c$ $(a\neq 0)$ satisfies $f(0)=-1$, for any $x\in\mathbb{R}$, we have $f(x)\geqslant x-1$, and $f\left(- \dfrac {1}{2}+x\right)=f\left(- \dfrac {1}{2}-x\right)$.  
$(1)$ Find the expression of the function $f(x)$;  
$(2)$ Does there exist a real number $a$ such that the function $g(x)=\log_{\frac {1}{2}}[f(a)]^{x}$ is decreasing on $(-\infty,+\infty)$? If it exists, find the range of the real number $a$; if not, explain why.

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