
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
A function $f(x)$ defined on a domain $D$ is said to be bounded if for any $x \in D$, there exists a constant $M > 0$ such that $|f(x)| \leqslant M$ holds. The constant $M$ is referred to as an upper bound of the function $f(x)$:
$(1)$ Given $f(x)= \dfrac {x}{x+1}$, assess whether $f(x)$ is a bounded function on the interval $[- \frac {1}{2}, \frac {1}{2}]$. If so, please explain the reason and write down the set of all possible upper bounds for $f(x)$. If not, also provide the reasoning.
$(2)$ If the function $g(x)=1+a\cdot \left( \frac {1}{2} \right)^{x}+\left( \frac {1}{4} \right)^{x}$ is bounded on the interval $[0,+\infty)$ with $3$ as an upper bound, determine the range of the real number $a$.

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