
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
Students, have you noticed: the spider silk drooping naturally with dewdrops after a rainy morning; the wires between two poles across the empty fields; the cables of the sightseeing ropeway across the canyon. These phenomena all share a similar curve shape. In fact, these curves are often referred to as catenary curves in mathematics. The theory related to catenary curves has a wide range of applications in engineering, navigation, optics, and other fields. Below, we will study a class of functions related to catenary curves, expressed as $f(x)=ae^{x}+be^{-x}$ where $a$, $b$ are non-zero constants, and the irrational number $e=2.71828…$.
1. When $a=1$, if $f(x)$ is an even function, then $b=$ ______ ;
2. If $f(x)$ is a monotonic function on $\mathbb{R}$, please write a set of values for $a$, $b$ that meet the condition;
3. If the minimum value of $f(x)$ is $2$, find the minimum value of $a+b$.

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