
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 distinct real numbers such that $a + b = k$ and $c + d = k^2$, where $k$ is a positive real number. Also, let the roots of $x^2 - 4ax - 5b = 0$ be $c$ and $d,$ and the roots of $x^2 - 4cx - 5d = 0$ be $a$ and $b.$ Find the value of $a + b + c + d.$

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