
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
Given plane vectors $\overrightarrow{a}=(x,y)$, $\overrightarrow{b}=(x^{2},y^{2})$, $\overrightarrow{c}=(1,1)$, $\overrightarrow{d}=(2,2)$, if $\overrightarrow{a} \cdot \overrightarrow{c} = \overrightarrow{b} \cdot \overrightarrow{d} = 1$, then how many such vectors $\overrightarrow{a}$ are there?

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