
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 the sum of all such integers \( a \in [0, 400] \) for which the equation

\[ x^{4} - 6x^{2} + 4 = \sin \left( \frac{\pi a}{200} \right) - 2\left\lfloor x^{2} \right\rfloor \]

has exactly six roots. Here, \(\left\lfloor t \right\rfloor\) denotes the greatest integer less than or equal to \( t \).

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