
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 that $y$ is a function of $x$, if the graph of the function passes through the point $P\left(t,t\right)$, then point $P$ is called a "balance point" on the graph of the function. For example, the line $y=-2x+3$ has a "balance point" $P\left(1,1\right)$. If the function $y=\left(m-1\right)x^{2}-3x+2m$ has a unique "balance point" on its graph, then $m=\_\_\_\_\_\_$.

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