
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
Curve $C$ is the locus of points in a plane such that the sum of its distances to three fixed points $F\_1(-1,0)$, $F\_2(1,0)$, and $F\_3(0,1)$ is equal to $2 \sqrt {2}$. Among the following conclusions:

$①$ Curve $C$ is symmetrical about both the $x$-axis and the $y$-axis;

$②$ There exists a point $P$ on curve $C$ such that $|PF_{3}|= \dfrac {2 \sqrt {2}}{3}$;

$③$ If point $P$ is on curve $C$, then the maximum area of $\triangle F_{1}PF_{2}$ is $1$;

$④$ The maximum area of $\triangle PF_{2}F_{3}$ is $\dfrac { \sqrt {3}}{2}$;

The sequence number(s) of all true propositions is/are _______ .

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