
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  $\mathcal S$  be the sphere with center  $(0,0,1)$  and radius  $1$  in  $\mathbb R^3$ .  A plane  $\mathcal P$  is tangent to  $\mathcal S$  at the point  $(x_0,y_0,z_0)$ , where  $x_0$ ,  $y_0$ , and  $z_0$  are all positive.  Suppose the intersection of plane  $\mathcal P$  with the  $xy$ -plane is the line with equation  $2x+y=10$  in  $xy$ -space.  What is  $z_0$ ?

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