
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
Given a finite sequence $T = (1, x, x^2, x^3, x^4)$, let $B(T)$ denote the sequence $\left(\frac{1+x}{2}, \frac{x+x^2}{2}, \frac{x^2+x^3}{2}, \frac{x^3+x^4}{2}\right)$. Define $B^1(T) = B(T)$ and for each integer $m$, $2 \le m \le 4$, define $B^m(T) = B(B^{m-1}(T))$. Suppose $x > 0$ and $B^4(T) = (\frac{1}{2^2})$. Determine $x$.
A) $1 - \frac{\sqrt{2}}{2}$
B) $\sqrt{2} - 1$
C) $1/\sqrt{2}$
D) $2 - \sqrt{2}$

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