
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 equation $\Gamma$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$, a line $l: x+y-4\sqrt{2}=0$, the lower end points of $\Gamma$ are $A$, $M$ is on $l$, and the left and right foci are $F_{1}(-\sqrt{2}$, $0)$, $F_{2}(\sqrt{2}$, $0)$.<br/>$(1)$ $a=2$, the midpoint of $AM$ is on the $x$-axis, find the coordinates of point $M$;<br/>$(2)$ Line $l$ intersects the $y$-axis at $B$, line $AM$ passes through the right focus $F_{2}$, and in $\triangle ABM$ there is an interior angle with cosine value of $\frac{3}{5}$, find $b$;<br/>$(3)$ There exists a point $P$ on the ellipse $\Gamma$ such that the distance from $P$ to $l$ is $d$, there exists a point on $\Gamma$ such that $|PF_{1}|+|PF_{2}|+d=6$, as $a$ varies, find the minimum value of $d$.

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