
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 base of the pyramid \( S A B C D \) is a quadrilateral \( A B C D \). The edge \( S D \) is the height of the pyramid. Find the volume of the pyramid, given that \( |A B| = |B C| = \sqrt{5} \), \( |A D| = |D C| = \sqrt{2} \), \( |A C| = 2 \), and \( |S A| + |S B| = 2 + \sqrt{5} \).

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