
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 vertex of the ellipse  $\frac{x^2}8+\frac{y^2}4 = 1$  in the Cartesian Plane. For some real number  $k$ , the line  $y=kx+1$  meets the ellipse at two distinct points  $A, B$ .

(i) Compute the maximum of  $|CA|+|CB|$ .
(ii) Let the line  $y=kx+1$  meet the  $x$  and  $y$  axes at  $M$  and  $N$ , respectively. If the intersection of the perpendicular bisector of  $MN$  and the circle with diameter  $MN$  lies inside the given ellipse, compute the range of possible values of  $k$ .

*(Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 12)*

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