
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
Let  $a,b,c\in \mathbb{R}$  with  $a^2+b^2+c^2=1$  and  $\lambda\in \mathbb{R}_{>0}\setminus\{1\}$ . Then for each solution  $(x,y,z)$  of the system of equations:
\[
\begin{cases}
x-\lambda y=a,
y-\lambda z=b,
z-\lambda x=c.
\end{cases}
\] 
we have  $\displaystyle x^2+y^2+z^2\leqslant \frac1{(\lambda-1)^2}$ .

*Radu Gologan*

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