
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with the right focus at $F(c,0)$, the vertex $B(0,b)$ of the ellipse, and the origin $O$ at the coordinate origin. It is known that $\angle FBO = \frac{\pi}{6}$ and the perimeter of $\triangle FBO$ is $3 + \sqrt{3}$. $P$ is a moving point on the ellipse, and $M$ is a point on the line $x = 4$ such that the line $PM$ is parallel to the $x$-axis.
1. Find the equation of the ellipse $C$.
2. Let the intersection of the line $PF$ with the ellipse be $Q$. Does the line $QM$ pass through a fixed point on the $x$-axis? If yes, find the coordinates of that fixed point. If not, explain why.

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