
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 the vector $\overrightarrow{d}=(1,-1)$ is perpendicular to the normal vector of the line $l$, perpendiculars are drawn from $A(1,1)$ and $B(-1,8)$ to line $l$, with the corresponding foot of the perpendiculars being $A_{1}$ and $B_{1}$. If $\overrightarrow{{A}_{1}{B}_{1}}=λ\overrightarrow{d}$, then the value of the real number $\lambda$ is ______.

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