
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
We constructed a symmetric trapezoid from two right triangles with sides of 3, 4, and 5 units and one rectangle with sides of 1 and 4 units.

Notably, the double area \(2t = 8 \cdot 4 = 32 \) square units, and if we add the square of the leg's length to the product of the parallel sides, we also get \(32 - t \). Thus, for our trapezoid:

\[ m(a+c) = b^2 + ac \]

It is also noticeable that:

\[ b^2 = \frac{7^2 + 1^2}{2} \]

that is, \( b \) is the quadratic mean of \( a \) and \( c \):

\[ b^2 = \frac{a^2 + c^2}{2} \]

Is it a coincidence that these two relationships hold simultaneously?

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