
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 $S = (1, x, x^2, \ldots, x^9)$ of $n=10$ real numbers, and the transformation $A(S)$ defined as $\left(\alpha\frac{a_1+a_2}{2}, \alpha\frac{a_2+a_3}{2}, \ldots, \alpha\frac{a_{9}+a_{10}}{2}\right)$, where $\alpha = \frac{1}{2}$, and for each integer $m$, $A^m(S)=A(A^{m-1}(S))$, find the value of $x$ if $A^{10}(S) = \left(\frac{1}{2^5}\right)$.

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