
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 hyperbola $C$: $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ $(a \gt 0, b \gt 0)$ with left and right foci $F_{1}$, $F_{2}$, and $|F_{1}F_{2}|=4$. A line $l:y=-\frac{\sqrt{3}}{3}x$ is perpendicular to one of the asymptotes of $C$. <br/>$(1)$ Find the standard equation of $C$;<br/>$(2)$ Let point $M$ be a moving point on $C$. Lines $MF_{1}$ and $MF_{2}$ intersect $C$ at two distinct points $A$ and $B$ (both different from $M$), and $\overrightarrow{MF_{1}}=\lambda\overrightarrow{F_{1}A}$, $\overrightarrow{MF_{2}}=\mu\overrightarrow{F_{2}B}$. Is $\lambda +\mu$ a constant value? If it is, find the value. If not, explain why.

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