
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 circle $B$: $(x-1)^{2}+(y-1)^{2}=2$, two different lines $l_{1}$ and $l_{2}$ pass through the origin $O$ and intersect with circle $B$.

$(1)$ Perpendicular lines are drawn from point $B$ to $l_{1}$ and $l_{2}$, with the feet of the perpendiculars being points $A$ and $C$, respectively. If $\overrightarrow{BA} \cdot \overrightarrow{BC}=0$ and $|\overrightarrow{BA}|=|\overrightarrow{BC}|$, find the equation of line $AC$;

$(2)$ If $l_{1} \perp l_{2}$, and $l_{1}$, $l_{2}$ intersect with circle $B$ at points $P$ and $Q$ respectively, find the maximum area of $\triangle OPQ$.

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