
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 $f(x)$ be a function defined on $R$ that satisfies the following two conditions:<br/>① For any $x\in R$, $f(x)=f(-x)-2x$;<br/>② For any $x_{1}$, $x_{2}\in [0,+\infty)$ and $x_{1}\neq x_{2}$, $(x_{1}-x_{2})[f(x_{1})-f(x_{2})] \gt 0$.<br/>Then the solution set of the inequality $f(2x+1)+x \gt f(x+1)$ is ______.

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