
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 $f(x) = e^{|x|} + a$ (where $e = 2.71828\ldots$ is the base of the natural logarithm), its minimum value is $1$.

$(1)$ Find the value of the real number $a$;

$(2)$ Given $b \in \mathbb{R}$ and $x < 0$, solve the inequality $\ln f(x) < x^2 + (2b-1)x - 3b^2$ with respect to $x$;

$(3)$ Given $m \in \mathbb{Z}$ and $m > 1$, if there exists a real number $t \in [-1, +\infty)$, such that for any $x \in [1, m]$, $f(x+t) \leqslant ex$, find the maximum value of $m$.

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