
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
Consider the equation of the parabola \( y = x^2 + 2x - 6 \). Let \( L(m) \) denote the \( x \)-coordinate of the left endpoint of the intersection of the graphs of \( y = x^2 + 2x - 6 \) and \( y = m \), where \( -8 < m < 8 \). Define \( r = \dfrac{L(-m) - L(m)}{m} \). Determine the behavior of \( r \) as \( m \) approaches zero.
A) $\frac{1}{\sqrt{5}}$
B) $\frac{1}{\sqrt{6}}$
C) $\frac{1}{\sqrt{7}}$
D) $\frac{1}{\sqrt{8}}$
E) $\frac{1}{\sqrt{9}}$

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