
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
Find all values of the parameter \(a\), for each of which the set of solutions to the inequality \(\frac{x^{2}+(a+1) x+a}{x^{2}+5 x+4} \geq 0\) is the union of three non-overlapping intervals. In your answer, specify the sum of the three smallest integer values of \(a\) from the obtained interval.

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