
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
For the equation $g(x) = t$ ($t \in \mathbb{R}$) regarding $x$, the number of real roots is denoted as $f(t)$. If $g(x) = \ln x$, then $f(t) = \_\_\_\_\_\_\_$; if $g(x) = \begin{cases} & x, & x\leqslant 0, \\ & -x^{2}+2ax+a, & x > 0, \end{cases}$ ($a\in \mathbb{R}$), and there exists $t$ such that $f(t+2) > f(t)$ holds, then the range of values for $a$ is $\_\_\_\_\_\_\_\_\_$.

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