
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
Natural numbers  $k, l,p$  and  $q$  are such that if  $a$  and  $b$  are roots of  $x^2 - kx + l = 0$  then  $a +\frac1b$  and  $b + \frac1a$  are the roots of  $x^2 -px + q = 0$ . What is the sum of all possible values of  $q$ ?

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