
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
Call a polynomial  $ P(x_{1}, \ldots, x_{k})$  *good* if there exist  $ 2\times 2$  real matrices  $ A_{1}, \ldots, A_{k}$  such that 
 $ P(x_{1}, \ldots, x_{k}) = \det \left(\sum_{i=1}^{k}x_{i}A_{i}\right).$  

Find all values of  $ k$  for which all homogeneous polynomials with  $ k$  variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)

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