
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
 $ABCD$  is a regular tetrahedron with side length 1. Points  $X,$   $Y,$  and  $Z,$  distinct from  $A,$   $B,$  and  $C,$  respectively, are drawn such that  $BCDX,$   $ACDY,$  and  $ABDZ$  are also regular tetrahedra. If the volume of the polyhedron with faces  $ABC,$   $XYZ,$   $BXC,$   $XCY,$   $CYA,$   $YAZ,$   $AZB,$  and  $ZBX$  can be written as  $\frac{a\sqrt{b}}{c}$  for positive integers  $a,b,c$  with  $\gcd(a,c) = 1$  and  $b$  squarefree, find  $a+b+c.$ *Proposed by Jason Wang*

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