
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  $A=\left(
\begin{array}{ccc}
1 & 1& 0 
0 & 1& 0 
0 &0 & 2 
\end{array}
\right),\ B=\left(
\begin{array}{ccc}
a & 1& 0 
b & 2& c 
0 &0 & a+1 
\end{array}
\right)\ (a,\ b,\ c\in{\mathbb{C}}).$ (1) Find the condition for  $a,\ b,\ c$  such that  ${\text{rank} (AB-BA})\leq 1.$ (2) Under the condition of (1), find the condition for  $a,\ b,\ c$  such that  $B$  is diagonalizable.

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