
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 the function $f(x) = \begin{cases} \frac{1}{x}-1, & x>1 \\ -2x+a, & x\leq 1 \end{cases}$ is defined on $\mathbb{R}$ and satisfies: for any $x_1 \neq x_2$, $f(x_1) \neq f(x_2)$, determine the range of the real number $a$.

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