
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
In the plane, two points $F_1$ and $F_2$ satisfy $|F_1F_2|=4$. Let $d$ be a real number, and let $D$ represent the set of all points $P$ in the plane that satisfy $||PF_1|-|PF_2||=d$. Also, let $C$ be the circle in the plane with center $F_1$ and radius $6$. Among the following statements, select the correct ones (write down the numbers of all correct statements).
① When $d=0$, $D$ is a straight line;
② When $d=1$, $D$ is a hyperbola;
③ When $d=2$, $D$ intersects circle $C$ at two points;
④ When $d=4$, $D$ intersects circle $C$ at four points;
⑤ When $d=4$, $D$ does not exist.

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