
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 smallest real constant \( C \) such that for all \( a_1, a_2, a_3, a_4, a_5 \) (not necessarily distinct), there always exist four distinct indices \( i, j, k, l \) such that:

\[ 
\left| \frac{a_i}{a_j} - \frac{a_k}{a_l} \right| \leq C 
\]

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