
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 area of the larger base of a truncated pyramid is $T$, the area of the smaller base is $t$, and the height is $m$. The volume is $V = kTm$, where $\frac{1}{3} \leq k \leq 1$. What is the proportionality ratio $\lambda(<1)$ between the two bases? (The proportionality ratio is the ratio of corresponding distances in similar shapes.)

Application:
a) The area of the larger base of a truncated pyramid is $T = 532.8 \text{ cm}^2$, and the height is $m = 23.2 \text{ cm}$. The ratio of the corresponding sides of the bases is $3:5$. What is the volume?

b) The volume of a truncated pyramid is half that of a prism whose base is congruent to the larger base of the truncated pyramid and whose height is the same as the height of the truncated pyramid. What is the ratio of the areas of the two bases?

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