
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 that $O$ is the coordinate origin, the line $l_{1}$ is $x+y-4=0$, and the line $l_{2}$ is $y=2x$. $l_{1}$ and $l_{2}$ intersect at point $A$. <br/>$(1)$ Find the coordinates of point $A$; <br/>$(2)$ If point $B$ lies on $l_{1}$ and $OA \perp OB$, find the length of segment $AB$.

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