
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
On the lateral edges \( SA \), \( SB \), and \( SC \) of a quadrilateral pyramid \( SABCD \), whose base is a square, points \( A_1 \), \( B_1 \), and \( C_1 \) are taken such that \( SA_1 : SA = 3 : 7 \), \( SB_1 : SB = 2 : 7 \), and \( SC_1 : SC = 4 : 9 \). A plane passing through the points \( A_1 \), \( B_1 \), and \( C_1 \) intersects edge \( SD \) at point \( D_1 \). Find the ratio \( SD_1 : SD \) and the ratio of the volume of pyramid \( SA_1B_1C_1D_1 \) to the volume of pyramid \( SABCD \).

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