This is easy to track down using a voltmeter. While 'starting', measure voltage from battery post to post. If it stays high, e.g.12V, battery is good. Measure from battery post clamp to clamp, then proceed down the line. Common problems are poor contact at battery posts, or inside solenoid relay.

Steve

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'91 MB 190E 2.3
'08 RAV4 Ltd 3.5
'83 Lazy Daze m'home 5.7