
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
For every positive integer  $k$ , let  $\mathbf{T}_k = (k(k+1), 0)$ , and define  $\mathcal{H}_k$  as the homothety centered at  $\mathbf{T}_k$  with ratio  $\tfrac{1}{2}$  if  $k$  is odd and  $\tfrac{2}{3}$  is  $k$  is even. Suppose  $P = (x,y)$  is a point such that   $$ (\mathcal{H}_{4} \circ \mathcal{H}_{3} \circ \mathcal{H}_2 \circ \mathcal{H}_1)(P) = (20, 20). $$   What is  $x+y$ ?
(A *homothety*  $\mathcal{H}$  with nonzero ratio  $r$  centered at a point  $P$  maps each point  $X$  to the point  $Y$  on ray  $\overrightarrow{PX}$  such that  $PY = rPX$ .)

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