
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
In the  $xy$ -coordinate plane, the  $x$ -axis and the line  $y=x$  are mirrors.  If you shoot a laser beam from the point  $(126, 21)$  toward a point on the positive  $x$ -axis, there are  $3$  places you can aim at where the beam will bounce off the mirrors and eventually return to  $(126, 21)$ .  They are  $(126, 0)$ ,  $(105, 0)$ , and a third point  $(d, 0)$ .  What is  $d$ ?  (Recall that when light bounces off a mirror, the angle of incidence has the same measure as the angle of reflection.)

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