
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) Let there be two given points  $A,B$  in the plane.
i. Find the triangles  $MAB$  with the given area and the minimal perimeter.
ii. Find the triangles  $MAB$  with a given perimeter and the maximal area.
(b) In a tetrahedron of volume  $V$ , let  $a,b,c,d$  be the lengths of its four edges, no three of which are coplanar, and let  $L=a+b+c+d$ . Determine the maximum value of  $\frac V{L^3}$ .

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