
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 a circle $C_1$ with its center at the origin $O$ and tangent to the line $l_1: x - \sqrt{2}y + 6 = 0$, let $A$ be a variable point on the circle, with $AM \perp x$-axis at point $M$, and let $N$ be a point satisfying $\vec{ON} = \frac{1}{2} \vec{OA} + \left(\frac{\sqrt{3}}{3} - \frac{1}{2}\right) \vec{OM}$. Let the locus of point $N$ be denoted by curve $C$.
(1) Find the equation of curve $C$.
(2) If a variable line $l_2: y = kx + m$ intersects the curve $C$ at exactly one point, and perpendiculars $F_1P$ and $F_2Q$ are dropped from points $F_1(-1,0)$ and $F_2(1,0)$ to $l_2$ with feet $P$ and $Q$ respectively, and let $d_1$ be the distance from $F_1$ to the line $l_2$, $d_2$ be the distance from $F_2$ to the line $l_2$, and $d_3$ be the distance between points $P$ and $Q$, investigate if there is an extremal value of $(d_1 + d_2) \cdot d_3$. If so, find this extremal value.

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