
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 paper equilateral triangle of side length  $2$  on a table has vertices labeled  $A,B,C.$  Let  $M$  be the point on the sheet of paper halfway between  $A$  and  $C.$  Over time, point  $M$  is lifted upwards, folding the triangle along segment  $BM,$  while  $A,B,$  and  $C$  on the table. This continues until  $A$  and  $C$  touch. Find the maximum volume of tetrahedron  $ABCM$  at any time during this process.

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