
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
The equation of the chord of a circle can be found using the concept of "congruence." For example, given the circle $O: x^{2}+y^{2}=1$, two tangent lines are drawn through the point $P(-2,-2)$ to the circle $O$, with the points of tangency denoted as $A$ and $B$. The problem is to find the equation of the line $AB". The partial solution is as follows: Let $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$. From $\overrightarrow{PA} \cdot \overrightarrow{OA}=0$, simplification yields $x_1^2+y_1^2+2x_{1}+2y_{1}=0$. Since $x_1^2+y_1^2=1$, we have $2x_{1}+2y_{1}+1=0$. Similarly, we get $2x_{2}+2y_{2}+1=0$, and so on. Therefore, the equation of the line $AB$ is ______.

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