
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
Let  $a$  be real numbers. Given a matrix  $A=\left(\begin{array}{cc} a & 1-a  1-a & a \end{array}\right).$  Let  $\left(\begin{array}{c} x_0  y_0 \end{array}\right)=\left(\begin{array}{c} 1  0 \end{array}\right),$  and the sequences  $\{x_n\},\ \{y_n\}$  are defined by the following equation.

\[ \left(\begin{array}{c} x_n  y_n \end{array}\right)=A \left(\begin{array}{c} x_{n-1}  y_{n-1} \end{array}\right),\ \ n=1,2,\ \cdots \]

then find the necessray and sufficient condition for  $a$  such that the sequence  $\{x_n\}$  converges.

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