
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
For real non-zero numbers $a, b, c$ it is given that

$$
a^{2} - bc = b^{2} - ac = c^{2} - ab
$$

(a) What positive values can the expression

$$
\frac{a}{b+c} + \frac{2b}{a+c} + \frac{4c}{a+b}
$$

take? List all possible values. If the expression cannot take positive values, write 0.

(b) What negative values can the expression

$$
\frac{a}{b+c} + \frac{2b}{a+c} + \frac{4c}{a+b}
$$

take? List all possible values. If the expression cannot take negative values, write 0.

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