
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) In the Cartesian coordinate system $xOy$, if the line $y=2a$ has only one intersection point with the graph of the function $y=|x-a|-1$, then the value of $a$ is ______.

(2) Let $m\in\mathbb{R}$. A moving line passing through a fixed point $A$ is given by $x+my=0$, and another moving line passing through a fixed point $B$ is given by $mx-y-m+3=0$. They intersect at point $P(x,y)$. The maximum value of $|PA|\cdot|PB|$ is ______.

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