
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 circles $\odot O_{1}$ with radius $r_{1}$, and $\odot O_{2}$ with radius $r_{2}$, both externally tangent to a circle $\odot O$ with radius $r$ at points $A_{1}$ and $A_{2}$ respectively. When $r_{1}=1$, $r_{2}=2$, $r=3$, and the distance $A_{1} A_{2} = 4$, find the length of the common external tangent $T_{1} T_{2}$ between circles $\odot O_{1}$ and $\odot O_{2}$.

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