
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 point $G(2,2)$, two lines are drawn intersecting the parabola $y^{2}=2x$ at points $A$ and $B$. Let the slopes of lines $GA$ and $GB$ be $k_{1}$ and $k_{2}$, respectively. If $k_{1}+k_{2}=5$ and $k_{1}\cdot k_{2}=-2$, determine the equation of line $AB$.

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