
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 polynomials $R(x)$ of degree less than 4, for each of which there exists a polynomial $P(x)$ satisfying the identity $7 \sin^{31} t + 8 \sin^{18} t - 5 \sin^{5} t \cos^{4} t - 10 \sin^{2} t + 5 \sin^{5} t - 2 = P(\sin t) \left( \sin^{4} t - (1+\sin t) \left( \cos^{2} t - 2 \right) \right) + R(\sin t), \quad t \in \mathbb{R}$.

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