
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 $C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{3}}{2}$, and it passes through point $M\left( 4,1 \right)$.

(1) Find the equation of the ellipse $C$;

(2) If line $l:y=x+m(m\ne -3)$ intersects with ellipse $C$ at points $P,Q$, and the slopes of lines $MP,MQ$ are denoted as $\left({{k}_{1}},{{k}_{2}}\right)$, investigate whether $({{k}_{1}}+{{k}_{2}})$ is a constant value. If it is, find the value; if not, provide an explanation.

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