Typical input

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{{Search for|chicken lips}}

{{Search for/sandbox|chicken lips}}

Something that must be escaped

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{{Search for|1=E=MC²}}

{{Search for/sandbox|1=E=MC²}}

Test BASEPAGENAME as default

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Works as long as you don't force the first parameter input to be an explicit null (it just needs to be missing):

{{Search for}}

{{Search for/sandbox}}

{{Search for|1=}}

{{Search for/sandbox|1=}}

{{Search for|}}

{{Search for/sandbox|}}