
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) Let $T_{1}$ and $T_{2}$ be the boundaries of two equilateral triangles on a plane, positioned parallel to each other with a common center, and with side lengths of 1 and 2, respectively. What is the distance $\rho\left(T_{1}, T_{2}\right)$ from $T_{1}$ to $T_{2}$? What is the distance $\rho\left(T_{2}, T_{1}\right)$ from $T_{2}$ to $T_{1}$?

b) Let $K_{1}$ and $K_{2}$ be the boundaries of two cubes positioned parallel to each other with a common center and with edge lengths of 1 and 2, respectively. What is the distance $\rho\left(K_{1}, K_{2}\right)$ from $K_{1}$ to $K_{2}$? What is the distance $\rho\left(K_{2}, K_{1}\right)$ from $K_{2}$ to $K_{1}$?

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