
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
In the Cartesian coordinate plane $(xOy)$, a circle $(x-a)^2+(y-b)^2=1 (a,b \in \mathbb{R})$ intercepts the line $x+2y-1=0$ to form a chord of length $\frac{4}{5}\sqrt{5}$. Determine the maximum value of $ab$.

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