
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 two vectors  $v = (v_1,\dots,v_n)$  and  $w = (w_1\dots,w_n)$  in  $\mathbb{R}^n$ , lets define  $v*w$  as the matrix in which the element of row  $i$  and column  $j$  is  $v_iw_j$ . Supose that  $v$  and  $w$  are linearly independent. Find the rank of the matrix  $v*w - w*v.$ 

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