
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 two lines $L_1: x+y-1=0$ and $L_2: 2x-y+4=0$ intersecting at point P, and a variable line $L: ax-y-2a+1=0$:

(1) If the line L passes through point P, find the value of the real number $a$.
(2) If the line L is perpendicular to line $L_1$, calculate the area of the triangle formed by lines $L$, $L_1$, and $L_2$.

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