
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
There are 6 distinct values of  $x$  strictly between  $0$  and  $\frac{\pi}{2}$  that satisfy the equation 
\[ 
  \tan(15 x) = 15 \tan(x) . 
\]
Call these 6 values  $r_1$ ,  $r_2$ ,  $r_3$ ,  $r_4$ ,  $r_5$ , and  $r_6$ .  What is the value of the sum
\[
  \frac{1}{\tan^2 r_1} + 
   \frac{1}{\tan^2 r_2} + 
    \frac{1}{\tan^2 r_3} + 
     \frac{1}{\tan^2 r_4} + 
      \frac{1}{\tan^2 r_5} + 
       \frac{1}{\tan^2 r_6} \, ?
\]

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