
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 a quadratic function $f(x) = ax^2 + bx + c$ ($a > 0$, $c > 0$) whose graph intersects the x-axis at two distinct points, one of which is $(c, 0)$, and $f(x) > 0$ for $0 < x < c$.

(1) When $a=1$, $c= \frac {1}{2}$, find the solution to the inequality $f(x) < 0$;  
(2) Find the solution to the inequality $f(x) < 0$ (expressed in terms of $a$ and $c$);  
(3) If the area of the triangle formed by the three intersection points of the quadratic function's graph and the coordinate axes is 8, find the range of values for $a$;  
(4) If the inequality $m^2 - 2km + 1 + b + ac \geq 0$ holds true for all $k \in [-1, 1]$, find the range of values for the real number $m$.

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