
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 $f(x)=ax^{2}- \frac {1}{2}x+c$ $(a,c\in\mathbb{R})$ satisfies the conditions: $(1)f(1)=0$; $(2)$ For all $x\in\mathbb{R}$, $f(x)\geqslant 0$.

$(1)$ Find the values of $a$ and $c$:

$(2)$ Does there exist a real number $m$ such that the function $g(x)=f(x)-mx$ has a minimum value of $-5$ on the interval $[m,m+2]$? If it exists, find the value of the real number $m$; if not, explain why.

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