
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 two mirrors placed at a right angle to each other and two points A at  $ (x,y)$  and B at  $ (a,b)$ . Suppose a person standing at point A shines a laser pointer so that it hits both mirrors and then hits a person standing at point B (as shown in the picture). What is the total distance that the light ray travels, in terms of  $ a$ ,  $ b$ ,  $ x$ , and  $ y$ ? Assume that  $ x$ ,  $ y$ ,  $ a$ , and  $ b$  are positive. 

[asy]draw((0,4)--(0,0)--(4,0),linewidth(1));
draw((1,3)--(0,2),MidArrow);
draw((0,2)--(2,0),MidArrow);
draw((2,0)--(3,1),MidArrow);
dot((1,3));
dot((3,1));
label(" $A (x,y)$ ", (1,3),NE);
label(" $B (a,b)$ ", (3,1),NE);[/asy]

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