
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 functions $f\left(x\right)=ax-1$ and $g\left(x\right)=-x^{2}+2x+1$, if for any $x_{1}\in \left[-1,1\right]$, there exists $x_{2}\in \left[0,2\right]$ such that $f(x_{1}) \lt g(x_{2})$ holds, then the range of real number $a$ is ______.

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