
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 sequence $S = (1, x, x^2, \ldots, x^{50})$ of real numbers, define $A(S) = \left(\frac{a_1+a_2}{2}, \frac{a_2+a_3}{2}, \ldots, \frac{a_{49}+a_{50}{2}\right)$. Define further $A^1(S) = A(S)$ and for each integer $m$, $2 \le m \le 50$, $A^m(S) = A(A^{m-1}(S))$. If $x > 0$ and $A^{50}(S) = \left(\frac{1}{2^{25}}\right)$, what is $x$?
A) $\sqrt{2} - 1$
B) $1 - \sqrt{2}$
C) $\frac{1}{2}$
D) $2 - \sqrt{2}$
E) $\frac{\sqrt{2}}{2}$

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