
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 negative values of \( a \) for which the equation

$$
\frac{8 \pi a - \arcsin (\sin x) + 3 \arccos (\cos x) - a x}{3 + \operatorname{tg}^{2} x} = 0
$$

has exactly three solutions. Provide the sum of all found \( a \) (if no such \( a \) exist, indicate 0; if the sum of \( a \) is not an integer, round it to the nearest hundredth).

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