This is one area where Sheldon loses the plot in a serious way!
Fine.
Unless you're an extremely disciplined rider you will put more power in when climbing & you'll start to drift into anaerobic power production, once you've gone acidic it's only a matter of time before any bio-mechanial efficiency is completely wiped out by bio-chemical inefficiency. So you either slow down/stop or change the muscle groups producing the power to groups which aren't lactating & thus your climbing style. Generally this means if you're in the saddle standing up or if you're standing up dropping into the saddle, you can get more technical than this & break down the styles further but those are the 2 main styles.
When you get knackered sitting down try standing up for a bit and vice versa.. Otherwise just take a break.
Transferring this to the real world, when I was last in Italy climbing, so solo rides, if I just stayed in the saddle my HR would slowly rise & rise up to about 195bpm before I'd start to go into anaerobic inefficiency & need to slow down. For the same, or slightly higher, output if I alternated between sitting & standing I could keep my HR down to about 170-180bpm, the result was I could climb faster for longer. This applies not only when pushing hard but also on a relaxed ride.
As para above but with some empirical data.
To give you an idea how slowly you'd be going if you keep a consistent effort on a hill... say you're riding a long at a nice sedate 13mph on a city or trekking bike & you continue at that effort level as you ride up a 2.5% gradient you'll be doing about 7mph. As it gets steeper & gets to 5% you'll be doing about 4.5mph. With a 700x35c tyre 11-28 cassette you'll be looking at a cadence in the low 50rpm range for a 30/39/50 triple, if you go for a trekking 24/36/46 you'll be looking at 65rpm. Most of back roads over the rollers around cambridge have gradients in the 5% area & I wouldn't call this area exactly lumpy.
You need more effort to go up hill or you'll slow down. If you change down, you'll need to pedal faster to go as fast. Cambridgeshire really is pretty flat.