
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 largest real number  $\lambda$  with the following property: for any positive real numbers  $p,q,r,s$  there exists a complex number  $z=a+bi$ ( $a,b\in \mathbb{R})$   such that   $$  |b|\ge \lambda |a| \quad \text{and} \quad (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0. $$  

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