
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 \(a, b, c, d\) be nonnegative real numbers such that \(a + b + c + d = 1\).  Find the maximum value of

\[
\frac{ab}{a+b} + \frac{ac}{a+c} + \frac{ad}{a+d} + \frac{bc}{b+c} + \frac{bd}{b+d} + \frac{cd}{c+d}.
\]

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