
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
Define \( f(a) \) as a function representing the number of distinct solutions to the equation

$$
\sin \frac{a \pi x}{x^{2}+1}+\cos \frac{\pi\left(x^{2}+4 a x+1\right)}{4 x^{2}+4}=\sqrt{2-\sqrt{2}}
$$

For example, \( f(a_{1})=3 \) means that for \( a=a_{1} \), this equation has three distinct solutions \( x_{1}, x_{2}, x_{3} \). If the equation has no roots for \( a=a_{0} \), then \( f(a_{0})=0 \).

Solve the inequality \( f(a) \leq 5 \). Write the total length of the resulting intervals, rounding it, if necessary, to two decimal places. If there are no solutions, write the number 0; if the resulting length is infinite, write the number 9999.

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