
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
The set of quadruples  $(a,b,c,d)$  where each of  $a,b,c,d$  is either  $0$  or  $1$  is *called vertices of the four dimensional unit cube* or *4-cube* for short. Two vertices are called *adjacent*, if their respective quadruples differ by one variable only. Each two adjacent vertices are connected by an edge. A robot is moving through the edges of the 4-cube starting from  $(0,0,0,0)$  and each turn consists of passing an edge and moving to adjacent vertex. In how many ways can the robot go back to  $(0,0,0,0)$  after  $4042$  turns? Note that it is <u>NOT</u> forbidden for the robot to pass through  $(0,0,0,0)$  before the  $4042$ -nd turn.

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