
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
Find the remainder upon division by  $x^2-1$  of the determinant
  $$ \begin{vmatrix} 
x^3+3x & 2 & 1 & 0
 x^2+5x & 3 & 0 & 2
x^4 +x^2+1 & 2  & 1 & 3
x^5 +1 & 1 & 2  & 3
 \end{vmatrix} $$  

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