
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
Select points  $T_1,T_2$  and  $T_3$  in  $\mathbb{R}^3$  such that  $T_1=(0,1,0)$ ,  $T_2$  is at the origin, and  $T_3=(1,0,0)$ . Let  $T_0$  be a point on the line  $x=y=0$  with  $T_0\neq T_2$ . Suppose there exists a point  $X$  in the plane of  $\triangle T_1T_2T_3$  such that the quantity  $(XT_i)[T_{i+1}T_{i+2}T_{i+3}]$  is constant for all  $i=0$  to  $i=3$ , where  $[\mathcal{P}]$  denotes area of the polygon  $\mathcal{P}$  and indices are taken modulo 4.  What is the magnitude of the  $z$ -coordinate of  $T_0$ ?

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