
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 a circle $C: (x-1)^{2}+y^{2}=4$, two lines $l_{1}$ and $l_{2}$ passing through point $A(0,1)$ are perpendicular to each other. The distances from the center $C$ of the circle to the lines $l_{1}$ and $l_{2}$ are $d_{1}$ and $d_{2}$, respectively. Calculate the maximum value of $d_{1}d_{2}$.

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