
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 $M_{a}$ be the set of functions $f(x)$ such that there exists a positive real number $a$ for which $f(x+a) > f(x)$ for any $x$ in the domain of $f(x)$.
1. If $f(x)=2^{x}-x^{2}$, determine whether $f(x)$ is an element of $M_{1}$, and explain the reason;
2. If $g(x)=x^{3}- \dfrac {1}{4}x+3$, and $g(x) \in M_{a}$, find the range of values for $a$;
3. If $h(x)=\log _{3}(x+ \dfrac {k}{x}),\ x\in [1,+\infty)$ (where $k\in \mathbb{R}$), and $h(x) \in M_{2}$, find the minimum value of $h(x)$.

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