
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
Definition: The function $y=[x]$ is called the "floor function", where $[x]$ denotes the largest integer not greater than $x$; the function $y=\langle x \rangle$ is called the "ceiling function", where $\langle x \rangle$ denotes the smallest integer not less than $x$; for example, according to the definition, we have: $[1.3]=1$, $[-1.3]=-2$, $\langle -2.3 \rangle=-2$, $\langle 2.3 \rangle=3$  
(1) For the function $f(x)=\langle x \cdot [x] \rangle$, $x \in [-2, 2]$; find $f(- \frac {3}{2})$ and $f( \frac {3}{2})$;  
(2) Determine the parity of the function $f(x)$ in (1);  
(3) Express the function $y=[x]+\langle x \rangle$ for $(-1 \leq x \leq 1)$ in the form of a piecewise function.

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