
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
Let the linear functions of $x$ be $y=a_1x+b_1$ and $y=a_2x+b_2$. Then, the function $y=m(a_1x+b_1)+n(a_2x+b_2)$ (where $m+n=1$) is called the generating function of these two functions.  
(1) When $x=1$, find the value of the generating function of $y=x+1$ and $y=2x$;  
(2) If the intersection point of the graphs of $y=a_1x+b_1$ and $y=a_2x+b_2$ is $P$, determine whether point $P$ is on the graph of the generating function of these two functions and explain why.

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