
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 the line $l$: $y=kx+4(k\neq ±4)$ intersects the hyperbola $C$: $x^{2}- \frac{y^{2}}{3}=1$ at points $A$ and $B$, intersects the $x$-axis at point $Q$, and intersects the $y$-axis at point $P$. If $\overrightarrow{PQ}=λ_{1} \overrightarrow{QA}=λ_{2} \overrightarrow{QB}$, and $λ+λ_{2}=- \frac{8}{3}$, find $k^{2}=$ _____.

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