
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 $f(1) = 0$, and for any real number $x$, $f(x) \geq 0$.

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

(2) Does there exist a real number $m$ such that the function $g(x) = 4f(x) - mx$ has a minimum value of $-5$ in the interval $[m, m+2]$? If it exists, find the value of $m$; if not, explain why.

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