
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
For every pair of reals  $0 < a < b < 1$ , we define sequences  $\{x_n\}_{n \ge 0}$  and  $\{y_n\}_{n \ge 0}$  by  $x_0 = 0$ ,  $y_0 = 1$ , and for each integer  $n \ge 1$ : \begin{align*}
   x_n & = (1 - a) x_{n - 1} + a y_{n - 1},
    y_n & = (1 - b) x_{n - 1} + b y_{n - 1}.
\end{align*} The *supermean* of  $a$  and  $b$  is the limit of  $\{x_n\}$  as  $n$  approaches infinity. Over all pairs of real numbers  $(p, q)$  satisfying  $\left (p - \textstyle\frac{1}{2} \right)^2 + \left (q - \textstyle\frac{1}{2} \right)^2 \le \left(\textstyle\frac{1}{10}\right)^2$ , the minimum possible value of the supermean of  $p$  and  $q$  can be expressed as  $\textstyle\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $100m + n$ .

*Proposed by Lewis Chen*

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