
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
Introduce a standard scalar product in  $\mathbb{R}^4.$  Let  $V$  be a partial vector space in  $\mathbb{R}^4$  produced by  $\left(
\begin{array}{c}
1  
-1  
-1  
1 
\end{array}
\right),\left(
 \begin{array}{c}
1  -1  
1  
-1 
\end{array}
\right).$ Find a pair of base of orthogonal complement  $W$  for  $V$  in  $\mathbb{R}^4.$ 

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