
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 the function $f(x)=ax^{3}+ \frac{1}{2}(\sin θ)x^{2}-2x+c$ whose graph passes through the point $(1, \frac{37}{6})$, and is decreasing on $[-2,1]$ and increasing on $[1,+∞)$.
(I) Find the analytical expression of $f(x)$;
(II) If for any $x_{1}$, $x_{2}∈[m,m+3]$ $(m\geqslant 0)$, the inequality $|f(x_{1})-f(x_{2})|\leqslant \frac {45}{2}$ always holds, determine whether such an $m$ exists? If it does, find the range of values for $m$; if not, explain the reason.

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