
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 parabolas $y = (x - 2)^2$ and $x + 6 = (y - 5)^2$ intersect at four points $(x_1,y_1),$ $(x_2,y_2),$ $(x_3,y_3),$ and $(x_4,y_4).$  Find
\[
x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.
\]

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