
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
(1) If $\tan \alpha =3$, then the value of $\frac{\sin 2\alpha }{\cos^2 \alpha }$ is ______.

(2) Given that the line $y=2x+1$ is tangent to the curve $y=x^3+ax+b$ at the point $(1,3)$, then the value of the real number $b$ is ______.

(3) It is known that the line passing through the fixed point $A$ with equation $x+my=0$ and the line passing through the fixed point $B$ with equation $mx-y-m+3=0$ intersect at point $P$, then the maximum value of $|PA|+|PB|$ is ______.

(4) Given the function $f(x)=1-\frac{m e^x}{x^2+x+1}$, if there exists a unique positive integer $x_0$ such that $f(x_0) \geqslant 0$, then the range of the real number $m$ is ______.

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