
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 an ellipse with foci on the $x$-axis and an eccentricity that is the reciprocal of the eccentricity of the hyperbola $3x^{2}-y^{2}=3$, and passing through the point $(1, \frac{3}{2})$.
(1) Find the equation of the ellipse;
(2) If line $l$: $y=kx+m(k\neq 0)$ intersects the ellipse at two distinct points $M$, $N$, and point $P(\frac{1}{5},0)$ satisfies $|MP|=|NP|$, find the range of values of $k$.

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