
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
Terry the Tiger lives on a cube-shaped world with edge length  $2$ . Thus he walks on the outer surface. He is tied, with a leash of length  $2$ , to a post located at the center of one of the faces of the cube. The surface area of the region that Terry can roam on the cube can be represented as  $\frac{p \pi}{q} + a\sqrt{b}+c$  for integers  $a, b, c, p, q$  where no integer square greater than  $1$  divides  $b, p$  and  $q$  are coprime, and  $q > 0$ . What is  $p + q + a + b + c$ ? (Terry can be at a location if the shortest distance along the surface of the cube between that point and the post is less than or equal to  $2$ .)


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