
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 $S=(a_1,a_2,\ldots,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence $\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{n-1}+a_n}{2}\right)$ of $n-1$ real numbers. Define $A^1(S)=A(S)$ and for each integer $m$, $2\le m\le n-1$, define $A^m(S)=A(A^{m-1}(S))$. Assuming $x>0$, and letting $S=(1,x,x^2,\ldots ,x^{50})$. If $A^{50}(S)=(1/2^{25})$, then what is $x$?
A) $1 - \frac{\sqrt{2}}{2}$
B) $\sqrt{2} - 1$
C) $\frac{1}{2}$
D) $2 - \sqrt{2}$
E) $\frac{\sqrt{2}}{2}$

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