
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 the greatest real number \( a \) such that the inequality

\[ 
x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2} \geq a (x_{1} x_{2} + x_{2} x_{3} + x_{3} x_{4} + x_{4} x_{5})
\]

holds for every five real numbers \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \).

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