
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 base of the right triangular prism \( ABC A_1 B_1 C_1 \) is an isosceles right triangle with legs \( AC = BC = a \). The vertices \( M \) and \( N \) of a regular tetrahedron \( MNPQ \) lie on the line \( CA_1 \), and the vertices \( P \) and \( Q \) lie on the line \( AB_1 \). Find: a) the volume of the prism; b) the distance between the midpoints of segments \( MN \) and \( PQ \).

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