
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
A  $\emph{luns}$  with vertices  $X$  and  $Y$  is a region bounded by two circular arcs meeting at the endpoints  $X$  and  $Y$ . Let  $A$ ,  $B$ , and  $V$  be points such that  $\angle AVB=75^\circ$ ,  $AV=\sqrt{2}$  and  $BV=\sqrt{3}$ . Let  $\mathcal{L}$  be the largest area luns with vertices  $A$  and  $B$  that does not intersect the lines  $VA$  or  $VB$  in any points other than  $A$  and  $B$ . Define  $k$  as the area of  $\mathcal{L}$ . Find the value \[ \dfrac {k}{(1+\sqrt{3})^2}. \]

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