
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
Let $f(x) = x^3 + 2x^2 + 3x + 1$ and let $T$ be the set of integers $\{0, 1, 2, \dots, 30\}$. Determine how many integers $t$ in the set $T$ such that $f(t)$ has a remainder zero when divided by $5$.
A) 5
B) 6
C) 7
D) 8
E) 9

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