
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
Let  $ABCD$  be a square of side  $1$  and  $A'B'C'D'$  its image by a rotation  $r$  of center  $D$  . The intersection of these two squares is the quadrilateral  $DALC'$ . By  $C'$ , we draw the parallel to  $AD$ , which intersects  $AB$  in  $E$  .
(a) What is the area of the quadrilateral  $DALC'$  if  $E$  is the midpoint of  $[AB]$ ?
(b) Find the position of  $E$  in  $AB$  so that the area of  $DALC'$  is half the area of the initial square.
(c) Determine the position of  $E$  in  $AB$  for the area  $DALC'$  to be  $1/3 ,1/4, ...,1/m$ , the area of the initial square ( $m$  being a nonzero natural). 
![Image](https://1.bp.blogspot.com/-tk2eTvk1-8I/XzAPX_jf3KI/AAAAAAAAMTc/oOUALyAAc2Una9yjax_BA8P1Mdd87gMHACLcBGAsYHQ/s0/OBM%2B2013%2Bp3.jpg)

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