
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 $\Gamma$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with its left focus at $F$, its top vertex at $A$, and its major axis length as $2\sqrt{6}$. Let $B$ be a moving point on the line $l$: $x = -3$, and $M(m,0)$, with $AM \perp BM$. When $AB \perp l$, point $M$ coincides with $F$.
1. Find the equation of the ellipse $\Gamma$;
2. If the line $BM$ intersects the ellipse $\Gamma$ at points $P$ and $Q$, and if $AP \perp AQ$, find the value of $m$.

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