
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 polynomial equations:
\[100x^4 + ax^3 + bx^2 + cx + 25 = 0\]
and
\[25x^5 + dx^4 + ex^3 + fx^2 + gx + 100 = 0\]
These equations have a common rational root \( k \), which is not an integer and is negative. Determine \( k \).

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