
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
Determine all pairs  $(a, b)$  of real numbers for which there exists a unique symmetric  $2\times 2$  matrix  $M$  with real entries satisfying  $\mathrm{trace}(M)=a$  and  $\mathrm{det}(M)=b$ .

(Proposed by Stephan Wagner, Stellenbosch University)

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