
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
Find all positive values of \( a \) for which the equation

\[
\frac{2 \pi a+\arcsin (\sin x)+2 \arccos (\cos x)-a x}{\operatorname{tg}^{2} x+1}=0
\]

has exactly three distinct solutions within the interval \( (-\infty ; 7 \pi] \). Provide the sum of all such \( a \). If no such \( a \) exists, state 0; if \( a \) is not an integer, round it to the nearest hundredth.

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