
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
The equations
\[45x^4 + ax^3 + bx^2 + cx + 18 = 0\] and
\[18x^5 + dx^4 + ex^3 + fx^2 + gx + 45 = 0\]
have a common rational root $k$ which is not an integer, and which is negative. What is $k?$

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