
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 real numbers \( a \) and \( b \) satisfy the conditions
\[
\left\{\begin{array}{l}
a + b - 2 \geqslant 0 \\
b - a - 1 \leqslant 0 \\
a \leqslant 1
\end{array}\right.
\]
find the maximum value of \(\frac{a + 2b}{2a + b}\).

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