
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

Let $C$ be the left focus of the ellipse $\frac{x^{2}}{8}+\frac{y^{2}}{4}=1$, and the line $y=k x+1$ intersects the ellipse at points $A$ and $B$.

1. Find the maximum value of $|CA| + |CB|$.
2. If the line $y=k x+1$ intersects the $x$-axis and $y$-axis at points $M$ and $N$ respectively, and the circle with $MN$ as the diameter intersects the perpendicular bisector of segment $MN$ inside the ellipse (including its boundary), determine the range of values for the real number $k$.

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