
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 $f(x)= \begin{cases} x^{2}-4x+3, & \text{if } x\leqslant 0 \\ -x^{2}-2x+3, & \text{if } x > 0 \end{cases}$, the inequality $f(x+a) > f(2a-x)$ holds true for all $x$ in the interval $[a,a+1]$. Determine the range of real numbers $a$.

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