
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 that there are two fixed points $M(\sqrt{2},1)$ and $N(-\sqrt{2},-1)$ on the conic curve $E$, and $P$ is a moving point on the curve $E$ different from $M$ and $N$. When the slopes $k_{PM}$ and $k_{PN}$ of the lines $PM$ and $PN$ exist, it is known that ${k}_{PM}•{k}_{PN}=-\frac{1}{2}$.  
$(1)$ Find the standard equation of the conic curve $E$;  
$(2)$ If the line $l$: $x=my+\sqrt{2}$ intersects the conic curve $E$ at points $A$ and $B$, and intersects the $x$-axis at point $F$, and the projections of points $A$, $F$, $B$ on the line ${l'}$: $x=2\sqrt{2}$ are points $D$, $K$, $G$ respectively.  
① If the line $l$ intersects the $y$-axis at point $T$, and $\overrightarrow{TA}={λ}_{1}\overrightarrow{AF}$, $\overrightarrow{TB}={λ}_{2}\overrightarrow{BF}$, investigate whether the value of $\lambda _{1}+\lambda _{2}$ is constant as $m$ varies. If so, find the value of $\lambda _{1}+\lambda _{2}$; otherwise, explain the reason;  
② Connect $AG$ and $BD$, and investigate whether the lines $AG$ and $BD$ intersect at a fixed point as $m$ varies. If so, find the coordinates of the fixed point and provide a proof; otherwise, explain the reason.

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