This JavaScript will move the Action button result to a field in the form you can hide but save the value:

LFForm.onFormSubmission(async function (event) {
    const actionValue = event?.data?.action?.value;

    if (actionValue) {
        LFForm.setFieldValues({ fieldId: 5 }, actionValue);

        // Wait briefly to ensure the value is stored before submission
        await new Promise(resolve => setTimeout(resolve, 100));
    }
});

Change the field ID to the field ID in your form where you want to store the result.  I tested and it will work on first submission and task action buttons.