
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
Given a cube \(ABCD - A_1B_1C_1D_1\) with edge length 1, a sphere is constructed with center at vertex \(A\) and radius \(\frac{2 \sqrt{3}}{3}\). Find the length of the curve formed by the intersection of the sphere's surface with the cube's surface.

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