
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  $4$ -dimensional hypercube of edge length  $1$  is constructed in  $4$ -space with its edges parallel to
the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given
by  $(a, b, c, d)$ , where each of  $a, b, c,$  and  $d$  is either  $0$  or  $1$ . The  $3$ -dimensional hyperplane given
by  $x + y + z + w = 2$  intersects the hypercube at  $6$  of its vertices. Compute the  $3$ -dimensional
volume of the solid formed by the intersection.

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