
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
Given a polynomial \( R(x) \) of degree less than 4, and a polynomial \( P(x) \) such that:

\[ 
7 \sin^{31} t + 8 \sin^{13} t - 5 \sin^{5} t \cos^{4} t - 10 \sin^{7} t + 5 \sin^{5} t - 2 
\equiv P(\sin t)\left[\sin^{4} t - (1+\sin t)\left(\cos^{2} t - 2\right)\right] + R(\sin t), 
\]

where \( t \in \mathbb{R} \), find all such \( R(x) \).

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