
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
A ten-level  $2$ -tree is drawn in the plane: a vertex  $A_1$  is marked, it is connected by segments with two vertices  $B_1$  and  $B_2$ , each of  $B_1$  and  $B_2$  is connected by segments with two of the four vertices  $C_1, C_2, C_3, C_4$  (each  $C_i$  is connected with one  $B_j$  exactly); and so on, up to  $512$  vertices  $J_1, \ldots, J_{512}$ . Each of the vertices  $J_1, \ldots, J_{512}$  is coloured blue or golden. Consider all permutations  $f$  of the vertices of this tree, such that (i) if  $X$  and  $Y$  are connected with a segment, then so are  $f(X)$  and  $f(Y)$ , and (ii) if  $X$  is coloured, then  $f(X)$  has the same colour. Find the maximum  $M$  such that there are at least  $M$  permutations with these properties, regardless of the colouring.

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