
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
Consider the equations 
\[90x^4 + ax^3 + bx^2 + cx + 16 = 0\] 
and 
\[16x^5 + dx^4 + ex^3 + fx^2 + gx + 90 = 0\]
that have a common rational root $k$, which is not an integer and is negative. Identify $k$.

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