
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 $f(x)= \begin{cases} 1+ \frac {1}{x} & (x > 1) \\ x^{2}+1 & (-1\leqslant x\leqslant 1) \\ 2x+3 & (x < -1)\end{cases}$.
$(1)$ Find the value of $f\{f[f(-2)]\}$;
$(2)$ If $f(a)= \frac {3}{2}$, find the value of $a$.

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