
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
Suppose  $\mathcal{T}=A_0A_1A_2A_3$  is a  tetrahedron with  $\angle A_1A_0A_3 = \angle A_2A_0A_1 = \angle A_3A_0A_2 = 90^\circ$ ,  $A_0A_1=5, A_0A_2=12$  and  $A_0A_3=9$ . A cube  $A_0B_0C_0D_0E_0F_0G_0H_0$  with side length  $s$  is inscribed inside  $\mathcal{T}$  with  $B_0\in \overline{A_0A_1}, D_0 \in \overline{A_0A_2}, E_0 \in \overline{A_0A_3}$ , and  $G_0\in \triangle A_1A_2A_3$ ; what is  $s$ ?

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