
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
Given the straight line $2ax + (a + c)y + 2c = 0$ ($a \in \mathbb{R}$, $c \in \mathbb{R}$) passes through a fixed point $(m, n)$, and $x_{1} + x_{2} + m + n = 15$ with $x_{1} > x_{2}$, find the minimum value of $\frac{x_{1}^{2} + x_{2}^{2}}{x_{1} - x_{2}}$.

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