
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 $P$ be any point in the first quadrant on the hyperbola $\frac{x^{2}}{3}-y^{2}=1$. Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola, respectively. The lines $PF_{1}$ and $PF_{2}$ intersect the hyperbola at points $M$ and $N$ respectively. If $PF_{1}=\lambda_{1} \cdot F_{1}M$ and $PF_{2}=\lambda_{2} \cdot F_{2}N$, find the value of $\lambda_{1}+\lambda_{2}$ and the range of values for the slope $k_{MN}$ of the line $MN$.

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