
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:
$(p)$: The line segment with endpoints as the focus $F$ of the parabola $C$: $y^{2}=kx (k > 0)$ and the point $M(1, \sqrt {2})$ intersects with the parabola $C$.
$(q)$: The equation $\frac {x^{2}}{13-k^{2}}+ \frac {y^{2}}{2k-2}=1$ represents an ellipse with focus on the $x$-axis.
$(1)$ If $(q)$ is true, find the range of values for the real number $k$.
$(2)$ If $(p∧q)$ is false, $(p∨q)$ is true, find the range of values for the real number $k$.

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