
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
Let \( A \) and \( B \) be points in space for which \( AB = 1 \). Let \(\mathcal{R}\) be the region of points \( P \) for which \( AP \leq 1 \) and \( BP \leq 1 \). Compute the largest possible side length of a cube contained within \(\mathcal{R}\).

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