
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
For every  $x \ge -\frac{1}{e}\,$ , there is a unique number  $W(x) \ge -1$  such that 
\[
  W(x) e^{W(x)} = x.
\]
The function  $W$  is called Lambert's  $W$  function.  Let  $y$  be the unique positive number such that
\[
  \frac{y}{\log_{2} y} = - \frac{3}{5} \, .   
\]
The value of  $y$  is of the form  $e^{-W(z \ln 2)}$  for some rational number  $z$ .  What is the value of  $z$ ?

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