
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 real numbers  $a,\ b$ , define a point  $P_n(x_n,\ y_n)$  by 
\[(x_0,\ y_0)=(1,\ 0)\]

\[(x_{n+1},\ y_{n+1})=(ax_n-by_n,\ bx_n+ay_n)\ \ (n=0,\ 1,\ 2,\ \cdots).\]

Find all of  $(a,\ b)$  satisfying the following conditions (i) and (ii).

(i)  $P_0=P_6$ (ii) All of   $P_0,\ P_1,\ P_2,\ P_3,\ P_4,\ P_5$  are distinct.

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