
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 
\[
50x^4 + ax^3 + bx^2 + cx + 16 = 0
\]
and 
\[
16x^5 + dx^4 + ex^3 + fx^2 + gx + 50 = 0
\]
have a common rational root \( k \) which is negative. What is \( k \)?

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