
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
The side length of the lateral edge of a regular triangular prism $S-ABC$ is 2, with a base that is an equilateral triangle of side length 1. A cross-section passing through the side $AB$ divides the volume of the prism into two equal parts. What is the cosine of the dihedral angle between this cross-section and the base?

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