
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 the operation defined in $R$: $\begin{pmatrix}a & b \\ c & d\end{pmatrix} = ad - bc$, if the inequality $\begin{pmatrix}x-1 & a-2 \\ a+1 & x\end{pmatrix} \geq 1$ holds for any real number $x$, find the maximum value of the real number $a$.

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