
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
If the function $f(x)$ satisfies both conditions: ① $f(x)$ is an increasing function, ② $f(x+y)=f(x)f(y)$. Please write down a function that satisfies these conditions: $f(x)=\_\_\_\_\_\_$; if the proposition "$\exists x \gt 0$, the inequality $f(x)+2x+a \lt 0$ holds" is a false proposition, then the range of real number $a$ is ______.

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